Q:

Is 'his not having a name' grammatical correct?

As I read the following paragraph out of the book Breakfast at Tiffany's on page 39, 

She was still hugging the cat. "Poor slob," she said, tickling his
  head, "poor slob without a name."  "It's a little inconvenient, his
  not having a name. But I haven't any right to give him one: he'll
  have to wait until he belongs to somebody."

I can understand the meaning of the bold sentence, but I can not fully understand the usage of 'his'. From what I have searched, the meaning of his is this:

pronoun
  1. the possessive form of he1. (used as an attributive or predicative adjective): His coat is the brown one. This brown coat is his. Do you
  mind his speaking first?
  2. that or those belonging to him: His was the cleverest remark of all. I borrowed a tie of his.

It seems that the second meaning is the possible explanation, and from the sample sentences, I think 'his' here can be replaced by 'He' and 'him', is that right? 
What is the grammar rule for this usage of 'his'?

A:

Not having a name is a gerund. It is a verbal noun phrase. Since it is a noun phrase, it can be modified by a possessive pronoun such as his. And the possessive pronoun tells us whose not having a name we are talking about. In this case, it is his not having a name.
So, actually, his has its normal function of being a possessive pronoun. 
This page from the blog Get it Write talks about the use of "the possessive case" with gerunds (note that English does not have a full fledged "case system" like other  languages, so when we talk about "cases" in English, we need to do so with a grain of salt). At any rate, the blog does a good job of explaining why a possesive pronoun or other possessive construction (such as the woman's) is used before a gerund.
Here, Capote uses textbook grammar, based on the need to use the possessive his before a gerund. Note, however, that some writers might use him here instead of his, but that is less formal and many might consider it incorrect.
So, it is your reading this answer that I care about and which contains correct grammar, whereas you reading this answer is not grammatical according to this analysis. For more information, see this answer by StoneyB as to When to use an object pronoun or a possessive adjective before a gerund.

