I have been reflecting on the mighty Poly’s judgement that “a friend of him” is incorrect. (Dangerous word incorrect!)
Take the following sentence: “She was a friend of them both”. You will notice that this sounds fine; indeed, it sounds better than “… a friend of theirs both”. Okay, so this probably has something to do with the presence of the word “both”, whose function here is described by the Collins COBUILD Dictionary as an “emphasising pronoun”.
So, let’s delete the “both”, which gives us “She was a friend of them”. I think it can be argued that there is a potential difference in meaning between this and “I was a friend of theirs”. While the latter favours an interpretation whereby the friendly feelings belonged to “them”, the former perhaps (and especially in a careful speaker’s - think lawyer - mouth) supports an interpretation whereby the friendly feelings belonged to “her”. Now, such is the nature of friendship that it is normally assumed that feelings are held mutually on each side, but of course this needn’t be the case, and this is where I think the grammar can help us to differentiate.
Imagine the following courtroom exchange:
A: “He thought he was a friend.”
B: “A friend of who?” [hey - this is idiomatic! what else do you want? “of whom”? “of whose?”]
A: “A friend of me.”
A is able to disambiguate by using “of me” rather than “of mine”, to clarify that what he means is that he (A) never considered this fellow to be his friend. The friendly feelings belonged to him, were on his side, rather on the speaker’s.
So, substituting the third person pronoun for the first, it would be possible, and grammatical, to say “a friend of him”. Although, admittedly, most improbably in the sentence that Poly was referencing. However, if one were to hear an intelligent, articulate, grammatically- aware person, such as Poly, saying “I went to the movies with my cousin and a friend of him”, it would be unwise to rule out the possibility that he was exploiting the resources of the Englsih language to convey, very subtly, and yet very powerfully, his own opinion about the relations between his cousin and that other person!