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 answerthis answer by StoneyB as to When to use an object pronoun or a possessive adjective before a gerundWhen to use an object pronoun or a possessive adjective before a gerund.