Q:

Why do people still talk about bohmian mechanics/hidden variables

I was reading the Feynman lectures in physics and after thinking about it for a while it seems particularly unreasonable to talk about hidden variables. Let us say that the electron has some internal variables as yet unknown which determine its trajectory given a set of initial conditions just like in classical mechanics. But since these hidden variables are unobserved, coupling it with a classical system should make their effect unchanged. This is what Feynman says, I think, in the last paragraph of Ch1 Vol 3, that if in the double slit experiment, if these inner variables dictate that the electron goes through the upper slit and land at a particular place on the opposite screen, and some other place for the lower screen, then the probability must neccesarily be the sum of two Gaussian like peaks, which does not agree with experiment. 
So if I concluded that inner workings of an electron had some additional hidden variables, then it should yield, as they should be independent of the classical apparatus, mutually exclusive probabilities that do not quiet add up the way as observed. But then I do a hidden variables search on the archive and a lot of smart guys still write about it, as late as Feb 2011.
So the argument I have used might be somehow incomplete, can anyone explain how?
EDIT: Sorry for editing this question almost three years later. I tried to locate the exact reference from the Feynman lectures I was referring to and this is the updated source,
Sec 7 Ch 1 Vol 3 

We make now a few remarks on a suggestion that has sometimes been made
  to try to avoid the description we have given: “Perhaps the electron
  has some kind of internal works—some inner variables—that we do not
  yet know about. Perhaps that is why we cannot predict what will
  happen. If we could look more closely at the electron, we could be
  able to tell where it would end up.” So far as we know, that is
  impossible. We would still be in difficulty. Suppose we were to assume
  that inside the electron there is some kind of machinery that
  determines where it is going to end up. That machine must also
  determine which hole it is going to go through on its way. But we must
  not forget that what is inside the electron should not be dependent on
  what we do, and in particular upon whether we open or close one of the
  holes. So if an electron, before it starts, has already made up its
  mind (a) which hole it is going to use, and (b) where it is going to
  land, we should find P1 for those electrons that have chosen hole 1,
  P2 for those that have chosen hole 2, and necessarily the sum P1+P2
  for those that arrive through the two holes. There seems to be no way
  around this. But we have verified experimentally that that is not the
  case. And no one has figured a way out of this puzzle. So at the
  present time we must limit ourselves to computing probabilities. We
  say “at the present time,” but we suspect very strongly that it is
  something that will be with us forever—that it is impossible to beat
  that puzzle—that this is the way nature really is.

A:

I agree with Luboš that this question has a lot to do with psychology.
I think the tic-tac-toe analogy is relevant. There are an infinite number of games that are precisely equivalent to tic-tac-toe, but humans are probably terrible at playing most of these games.
Chess is even worse. You can teach a child the rules of chess in a few minutes, but imagine doing so without the board! Consider one of the simplest games of chess:

e4 e5
Qh5?! Nc6
Bc4 Nf6??
Qxf7# 1–0

A child can grasp the game, but even a grandmaster probably pictures a board as he reads those symbols.
A chess computer does not picture a chess board. It uses a different representation that is more efficient for its calculations. I suspect this formulation of chess would not hold a child's attention for very long, and only a very exceptional child would attain any skill at the game.
Some representations of chess or tic-tac-toe suit the human mind. Others are efficient to simulate on a computer. Still other representations are utterly useless, despite being precisely equivalent. Surely the same is true of the rules of physics.
For this reason, I like Bohm's efforts to find an experimentally indistinguishable reformulation of QM, whether or not he succeeded. Imagine if we've been playing 'JAM' all this time, when we could have been playing 'tic-tac-toe'!
I think that's why people still talk about Bohm.

A:

First point I want to address is a gross mischaracterization of people working on Bohmian mechanics or other hidden variable interpretations. I quote

Quantum mechanics remains unacceptable for many people, for quasi-religious reasons.
  [...]
  it's time for those people to seriously consider the hypothesis that quantum mechanics is really how the world works.
People for whom it is hard to swallow the hard reality of quantum mechanics, people who want to understand nature in terms of every day classical intuitions are the ones who often advocate Bohmian mechanics.

This is a straw man. Nobody I know working or having worked on hidden variable interpretations of quantum mechanics, be it Bell, Bohm, Goldstein, Bricmont, Maudlin, de Broglie or other, denies the reality of quantum mechanics. Quite the contrary, I would even say they are/were more aware of the implications than many people otherwise working with quantum mechanics. In particular they fully recognize the non-local nature of quantum mechanics. But I'll come back to that. I can of course not speak for ALL people dealing in hidden variable theories or other alternatives. I'm sure there will be a faire share of crackpots and other cranks. But there are many people who know their stuff.

Second point is the Zeilinger article that @Luboš Motl mentions. I've only skimmed through the article, but it seems that the only non-local hidden variable theories that are disposed of are ones devised by Leggett. Moreover, on page 3 of the article, the authors recognize that while their work indeed rules out a vast class of non-local hidden variable theories, it does not rule out Bohmian mechanics. So it is disingeneous to use this article to discredit Bohmian mechanics. The work in the article is however very valuable since it puts more strict constraints on the type of non-local hidden variable theories that can reproduce the results of quantum mechanics. This is an interesting question that is also discussed in the excellent book Quantum Paradoxes by Yakir Aharonov and Daniel Rohrlich. In one of the later chapters, they discuss the very special nature of non-locality in quantum mechanics, pointing out that there are many varying degrees of non-locality which are either stronger or weaker than the non-locality of quantum mechanics. They say it is still an open problem to find a precise characterization of the nature of quantum non-locality which could be turned into a postulate from which quantum mechanics could be derived in a way analoguous to the way special relativity is derived from its basic postulates.
Now, about Bell's theorem. I'll quote a piece of the following article which itself refers to a thought experiment proposed by Tim Maudlin. Of course, you'll recognize it is a variant of an EPR type experiment.

Here is a puzzle: two persons, call them $X$ and $Y$ , leave a room
  through opposite doors; at that point, each is asked a question. The
  precise nature of the questions does not matter, but there are three
  possible questions (say, $A$, $B$ and $C$). Each person must answer yes
  or no. This “experiment” is repeated many times, with sometimes
  the same question, sometimes diﬀerent questions being asked at the
  two doors. The two persons are allowed to decide, before leaving
  the room, to follow any strategy they want, but not to communicate
  with each other, after they have heard the questions.
  The statistics of answers have some strange properties. First
  of all, it turns out that when the two people are asked the same
  question, they always give the same answer. Is that mysterious? Of
  course not; they simply decide, before leaving the room, to follow a
  certain strategy: for example, to both say ‘yes’ if the question is $A$,
  ‘no’ if the question is $B$ and ‘no’ if the question is $C$. Altogether,
  there are $8 = 2^3$ different such strategies. Before proceeding further, the reader has to answer for himself or herself the following
  question: Is there any other way? Is there any way to account for
  the perfect correlations between the results at the two doors without assuming that the answers were predetermined (if we assume
  that the people cannot have any communication whatsoever with
  each other once the questions are asked)? I have never seen any
  suggestion of another possibility and I believe that if Bell’s theorem
  is arguably the most widely misunderstood result in the history of
  physics, it is precisely because this question is not answered before
  proceeding further.
So, let us consider, for the time being, the assumption that the
  answers are predetermined and let us call $v_i(\alpha) = \pm 1$, $i = X, Y$ ,
  $\alpha = A, B, C$ those answers. These are “random variables”, namely
  they may take diﬀerent values when one repeats the “experiment”.
  However, if one looks at the statistics of answers when diﬀerent
  questions are asked at the two doors, one ﬁnds that the frequencies
  of the events in which the same answers are given is 1/4. And this,
  combined with the perfect correlations is strange. Indeed, a version
  of the no hidden variable theorems (similar to the one discussed in
  the previous section), known as Bell’s theorem.

I'll leave out the proof, you can read it for yourself in the article. It is just the usual Bell type inequality.

What is the conclusion of all this? We started from one crucial
  assumption: absence of “communication” between the two persons
  once they are asked the questions. I will from now on revert to a
  less anthropomorphic language and call this assumption “locality”
  – assume that there is no causal connection whatsoever between the
  two wings of the experiment. Then, we are led to a contradiction,
  so that this assumption has to be dropped.
  It is important to understand the logic of the argument: the
  perfect correlations plus the absence of communication (i.e. locality) between the two wings of the experiments, leads us to postulate the existence of the variables $v_i(\alpha)$ [...]. However, merely assuming that
  those variables exist leads to a contradiction with the experimental
  results obtained when different questions are asked. To put it simply: locality plus perfect (anti)correlation implies hidden variables;
  however, the latter plus statistics when different angles are measured implies a contradiction. Both the perfect correlations and
  the statistics for diﬀerent angles are empirical results; the theorem
  is a theorem, namely a logical deduction; the only assumption was
  the lack of “communication”, or locality. Hence, locality has to be
  given up, period.

So, I'll stress again the most important point of this text: hidden variables are postulated to explain the correlations in the absence of communication (i.e. locality assumption). But this leads to a contradiction with the predictions of quantum mechanics. Quantum mechanics is right, we have tested it experimentally, therefore locality isn't right. I think that's fairly limpid. Anybody claiming that quantum mechanics doesn't force us to conclude non-locality is the one who doesn't really accept quantum mechanics, contrary to what detractors claimed in point 1.
In his book Speakable and unspeakable in quantum mechanics, which is really a collection of Bell's papers, there are various presentations of the theorem, some of which are clearer than others. Another book which does a good job at explaining the interpretation and consequences of Bell's theorem is Tim Maudlin's Quantum Non-Locality and Relativity.

And I've barely addressed Bohmian mechanics yet and the question of the OP. Why do people still work on Bohmian mechanics? Well, the original intent of EPR was to show that quantum mechanics is incomplete. EPR showed that assuming locality inevitably leads to hidden variables, which is the first part of the previous argument. But the second part of the argument, provided by Bell shows that this is inconsistent with quantum mechanics. Therefore locality can't be right. One might wonder, even if we have to drop locality, why insist on supplementing the theory with hidden variables?

It's important to understand that the only variables added in Bohm's theory are the positions of particles. There are no hidden variables for spin, momentum, angular momentum, etc... It does however explain how the measured values for spin, momentum, etc... arise from the specific experimental configurations. This is what is called contextuality and has already been mentioned by @Sina Salek. Non-locality is explained by the fact that the wave-function is a function on configuration space and not in physical space. Hence the possibility of entanglement. You can read more as well as find further references in the article I mentioned.
Beyond the fact that Bohmian mechanics makes the non-locality in quantum mechanics more explicit, it also provides a deterministic interpretation of quantum mechanics, showing that quantum mechanics does not force randomness upon us.
Now, wether one likes Bohmian mechanics or not, one can not deny that these are definitely strong points of the theory. However, it does have its weaknesses as well. As @sb1 mentioned, Einstein thought it was cheap and in a way, I do agree that the way the extra equation for the positions is added is rather cheap and even ugly. It does also have its fair share of problems, for instance in trying to generalize into a QFT.
Anyway, I leave this link containing a short exposition of Bohmian mechanics and further references.

Finally, I want to address @Matt Reece about QFT and locality. I'm no expert in QFT, but I do remember that locality is imposed on the level of operators by requiring that two observables localized within distinct space-time regions should commute. I'm not entirely sure what this implies w.r.t. entangled states, but I suppose it doesn't rule them out otherwise QFT would be in contradiction with experiment. (I wonder though if there is a complete treatment of entangled states within the context of QFT?) But if entangled states are allowed, then violations of Bell type inequalities still are possible and thus non-locality is a fact. What this means is that whatever the status of the principle of locality in QFT, it's a weaker form of locality than the one required in setting up Bell's inequalities.

A:

Bohmian mechanics is a non-local hidden variable theory. It can reproduce the results of ordinary quantum mechanics. Since Bell's theorem and the subsequent experimental verdict ruled out only local hidden variable theory, Bohmian mechanics can still survive in principle.
In reality however it survives due to a different reason as pointed out by Lubos. The reason is essentially regressive. People for whom it is hard to swallow the hard reality of quantum mechanics, people who want to understand nature in terms of every day classical intuitions are the ones who often advocate Bohmian mechanics. IMHO it is extremely difficult for a physicist to support this approach. Why? The reasons are as follows.

It violates Occam's razor by introducing extra equation in addition to the Schrödinger's equation.
It is explicitly nonlocal and inconsistent with special relativity.
It can not be extended to produce a quantum field theory.
It is an unnecessary superstructure over ordinary quantum mechanics.
The quantum mechanical spin becomes terribly messed up in this approach.

In short it is a cheap alternative to ordinary sane quantum mechanics.

