Do we say a friend of my uncle or a friend of my uncle's? In spite of the fact that a friend of my uncle's seems to overwork the notion of possessiveness, that is usually what we say and write.
The double possessive construction is sometimes called the post-genitive or of followed by a possessive case or an absolute possessive pronoun (from the Oxford English Dictionary which likes to show off).
The double possessive has been around since the fifteenth century, and is widely accepted. It's extremely helpful, for instance, in distinguishing between a picture of my father (in which we see the old man) and a picture of my father's (which he owns).
Native speakers will note how much more natural it is to say He's a fan of hers than he's a fan of her.
Generally, what follows the of in a double possessive will be definite and human, not otherwise, so we would say a friend of my uncle's but not a friend of the museum's [museum, instead].
What precedes the of is usually indefinite (a friend, not the best friend), unless it's preceded by the demonstratives this or that, as in this friend of my father's.