As this NGram shows, we nearly always use the possessive form of personal pronouns for friend of mine/his/ours/etc.
But when it comes to actual names, we prefer friend of Peter without the possessive apostrophe. That preference is even more marked with, say, friend of America. Not that I think the usage itself is particularly American – it's much the same with Britain.
Personally, I find friend of him grates. In general I've no strong feelings either way as to whether it's friend of Peter or friend of Peter's (though I deplore the possessive in this example), but in line with many others, I really don't like the possessive in relation to things like countries.
Why is this?
Edit: Noting an apparent "progression" (pronoun -> person -> nation) marked by reduction in use of the possessive, I checked at a finer "granularity". NGram shows that although it does occur, friend of me virtually "flatlines" against friend of mine. But the bias reduces through of you, of us, and by the time I get to of them it's much less extreme. There seems to be something "egocentric" about the double possessive.
Presumably when babies learn to speak, they soon notice that possessive pronouns, possessive apostrophes, and the word "of", all do the same job. Parents would correct a child who says "of mine's", but probably wouldn't even notice the same "redundancy" in "of Peter's". Younger speakers are unlikely to even be talking about something "of America's". Perhaps as we mature we tend to discard the "double possessive" for the more "distant" things that only adults are likely talk about, but we keep it for "closer" people because that's how we spoke when we were younger.
EDIT2 I note that I'm a great fan of him is vanishingly rare compared to …fan of his, but with …fan (of John) the double possessive occurs far less often than …friend (of John's). Usage seems to be affected by the noun before "of" as well as the one after it. This is getting complicated…
Best Answer
To me, "Friend of Peter" and "Friend of Peter's" mean the inverse of each other.
In "Joe is a friend of Peter", Joe is the active person in the friendship - it describes Joe's active relationship to Peter. Peter is one of the people Joe expresses friendship toward.
In "Joe is a friend of Peter's", Peter is the active person in the friendship - it describes Joe as being the object of Peter's friendship. Joe is one of the people Peter expresses friendship toward.
In most contexts, nothing is being implied about the inverse relationship, although friendship is usually reciprocal. It's usually more a matter of who the speaker knows about the relationship from. So if I'm introducing you to Joe, but we both know Peter, and Peter's talked about Joe, I might tell you that Joe is a friend of Peter's.
The distinction is probably most significant in high school. ;-)