Q:

How do you mentor undergraduate research?

Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that. 
There are two slightly more specific groups of questions I have about this process:

How would you go about choosing a problem? Specifically, should the student work on open problems or work through existing proofs? Are there lists of problems at that might be fruitful and approachable? 
What sort of guidance would you provide them? With what frequency would you meet? Would your meetings be closer to teaching or guiding them along on their own?

A:

As a former faculty member in several elite college and university departments, I have worked with and mentored a number of talented undergraduate and graduate students including several mathematics students.  The advice I give to a student in this regard depends on the student's interest, and greatly upon the student's career goals.  Students who pursue mathematics as a tool and firm background for a career in other technical disciplines (e.g., computer science, physics, electrical engineering), are most likely to profit from working on existing problems, even deep ones whose roots lie in other disciplines.  That's what their career will likely entail. 
If however the student seeks a career as a professional mathematician--particularly an academic, research or "pure" mathematician--the student must become expert in identifying and creating new problems.  For such students, I would counsel they immerse themselves in their favorite subdiscipline and spend some time creating the problem, or modifying a recognized one.
Being able to ask the right questions and identify new problems is a talent that distinguishes the greatest mathematicians (and scientists) and is, alas, rarely taught or even encouraged in most academic departments.  I published a peer-reviewed research paper with an extremely talented undergraduate mathematics student who solved a problem that I posed.  (I was not his senior thesis mentor, however.)  When he went off to one of the leading graduate mathematics departments he found that he could not identify new problems.  Last I heard he dropped out of graduate school and programmed computers for a traditional bank.

A:

I have only advised a few undergraduates on projects like this, although I participated in several undergraduate research projects including an REU and a baccalaureate thesis.
The choice between working on open problems versus reading existing proofs is a false dichotomy. 
First, it is common to have students do both: Have them read through papers you select and work on a problem. 
Second, those aren't the only two options. There are many things that are not known that are not called open problems. A student can collect data testing a conjecture, and this can range in difficulty from a straightforward calculation or programming exercise to something requiring real ingenuity to check examples with intrinsic interest. A student can apply a result to a case of interest.
There are many questions mathematicians ask that are not universally called open problems because people haven't looked at them much. This happens at the ends of many papers and many talks. The author of a paper might be saying, "I think the techniques used in this paper will work on the following problems but I don't have time to look at them." These are often much better problems for students to work on than problems that are known to be open because many people have verified that the easy/standard techniques don't work. 
Part of mathematics is exposition. You can ask a student to rewrite a known result using better notation or filling in the details. When a result was a culmination of multiple papers, it might be that the overall exposition is very different from what it should be. Fixing that might be a good task for a student... or it could be far too hard, and a good task for a team of seasoned mathematicians, as in the case of the classification of finite simple groups. Many masters theses are improved expositions of known results. I have found a lot of nice expositions written as senior theses.

Advising undergraduates properly can require a significant amount of preparation, and it should not be undertaken lightly. I think you should focus on the educational benefit to the students instead of the potential value of the research, so lean toward having the undergraduate read good papers and pick up useful ideas and skills over having the best chance to produce research. Further, the student is likely to run into obstacles and may be unable to produce new research within the short amount of time allowed. When that happens, will the student have something to write up anyway? Try to make a project with many ways to conclude well such as expositions and partial results. It is easier (but not necessarily easy) to schedule progress through experimental calculations, reading, and expositions than it is to plan on research progress. 

A:

"...should the student work on open problems...?"

I think that working on open problems is fine—even exciting—if you follow Pólya's
advice in
How to Solve It:

"If you can't solve a problem, then there is an easier problem you can solve: Find it."

Just as one example, it is unknown whether or not it is decidable if a given single polygonal tile can tile the plane. It is unknown even if the tile is a polyomino.
I supervised a very nice undergraduate exploration of this question for specific polyominoes.
If you simplify the question enough, you can solve it. Then generalize until you cannot solve it. And so on, oscillating above and below.

