Do we say "a friend of Joe's" or "a friend of Joe"?
In spite of the fact that "a friend of Joe'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 Joe'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."
That's all for now!