Q:

Why is it usually "friend of his", but no possessive apostrophe with "friend of Peter"? 

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...

A:

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. ;-) 

A:

I buried my lede way down below.  Headline summary: I believe the evidence shows that the choice is conditioned by a combination of semantic and syntactic criteria, namely animacy, definitiness, type of possession, and weight.  It doesn't appear to be a hard and fast rule, but rather a tendency to choose between forms based on the semantics.  It's very likely, therefore, that different speakers will make slightly different choices or have slightly different intuitions, but that we all agree on the "edge cases". 

This construction, often called the "double genitive", "double possessive", or "oblique: genitive, has a long history in English.  Compare:

"that hard heart of thine" (Shakespeare, Venus and Adonis, 1592)
"this extreme exactness of his" (Sterne, Tristram Shandy, 1759)
numerous uses can be found in Google Books even prior to 1600
compare "a picture of the king" vs "a picture of the king's"

So it's been around for a while, and long been remarked upon (the last reference above is from a phil0logical society proceedings in 1864.) 
The  salient point is that its usage is limited to personal referents.  
Personal referents would include not only the personal pronouns ("of mine", "of yours", etc.)  but proper and improper nouns referring to persons ("of John's", "of the king's").  I imagine that in personification it might be used by metaphor ("of America's").  But you would never have "of the door's" or the like; this explains the pattern seen by the questioner. 
This is evidently a result of its deeper origin as a partitive construction (by partitive I mean constructions like the modern "one of the soldiers" which is of similar origin).  It's not unusual for grammatical constructions to be conditioned by features like animacy, and for animacy to be heirarchical.  In English, animacy is a covert (semantic) category, so it can be elusive to notice.  

EDIT: Let me address what I believe is the crux of the question here, the "why" in Why is it usually “friend of his”, but no possessive apostrophe with “friend of Peter”? 
I will actually answer a slightly modified version of the question, to wit: 

Why is it almost invariably “friend of his”, but we sometimes find both “friend of Peter” or "friend of Peter's" in different
  circumstances? 

I will for now accept the stipulation that "friend of Peter" is found more often than "friend of Peter's". 
"Why" is always a difficult question, as there are no just-so-stories in linguistics.  From a purely empirical point of view, there are no whys. Examining the evidence, we find that the double genitive is used almost invariably in "friend of mine" and we almost never find "friend of me".  We likewise find that "friend of Peter" and "Peter's" are both found in distribution.  The "Peter's" construction is, however, marked.  It seems to be found more often in speech than in writing, and perhaps in certain dialects.  But none of this tells us "why", unfortunately. 
But we can look at the grammar of the construction.  Although this is more of a "how" than a "why", it might have some explanatory power.  Unfortunately the question here seems to be quite complex.  It's some combination of definiteness, proper vs. extrinsic possession, animacy, and weight.  Nearly all references agree that the double genitive is related to the partitive ("some of my friends") in its origin. 
My interpretation of all this is that when the noun in question is more definite, the possession is more proper, the animacy is high, and the weight is low, we are more likely to use "of x's".  
This paper at MIT has an excellent discussion of the questions surrounding this construction. 

A:

It’s misleading to think of the apostrophe as a possessive marker.  It’s more helpful to think of it as a genitive inflection, certainly capable of expressing possession (John’s car), but also used to specify or classify the reference of a noun (the girl’s face, a bird’s nest), to indicate time and place (a week’s holiday, the country’s capital) and to refer to a noun that is understood from the context (I’m going to a friend’s (house), Macy’s (store)).  Seen against that background, the use of the apostrophe after the name of a country is unexceptionable.  That’s one of Greece’s problems, for example, is surely more natural than That’s one of the problems of Greece. 