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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.

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.

Your primary question is 'why'. But first the phenomenon.

One says

to him

but

of his

All prepositions in English take the objective case, except for 'of' which takes the possessive, with pronouns.

One says

to Peter

and

of Peter

or

of Peter's

the unmarked case being more formal and the possessive more informal.

So that's just the phenomenon. Logically, one might think that 'of' should take the accusative pronoun to be consistent with other prepositions. Possibly the double genitive with pronouns is just a change by analogy with the phonetic idea of assimilation, two things close together modify one so they share a property.

As to the reason for the questioned phenomenon, since "of Peter's" is more informal and not standard, presumably it is caused by either grammatical assimilation just like the pronoun version, or it is by analogy -with- the pronoun version.

My guess would be that the friend of mine/his/ours constructions are simply idioms. Pronouns are far more restricted in their occurrences than are nouns, and occur in far more idioms than any noun can, just because Pronouns are a closed class. Pronouns have to fit the slots we need them for.

For instance, if you used of him or of them, you'd virtually have to contract either one in speech to "of'em", and the idiom requires a secondary stress on the preposition object, which contracting would lose. So friend of him doesn't sound right. So it's wrong.

As for NP vs NP's -- I have a vague supposition that logic may be winning this one. I'd like to have a dollar for every time somebody's asked me why we say friend of Bill's, when the possessive means the same as of, and we normally use an objective pronoun. There's no reason, I have to tell them; we just do. But people make up their own mind in the long run, and I think that intuition, which has occurred to every English speaker, may be telling. Especially in writing.

In speech, I would bet the proportions of friend of Jane's are higher than they are in the N-Gram. Writing, as we all know, falutes much higher than speech, and includes lots more strange stuff, so this is just more. Talking, on the other hand, is not rehearsed or organized in the same ways as writing, and is often more fond of familiar phrases than of clarity.

I had always imagined that the evolution of this originated in a slightly different meaning of of. If you interpret of in the sense of out of or among, then you can think of a friend of his/Peter's as a shorter form of a friend out of all of his/Peter's friends. Note that you are much less likely to say the friend of Peter's unless it is by contrast to some other group: the friend [I'm speaking of who is] among Peter's [friends] [rather than among John's].

Can anyone back up this interpretation with evidence in English or Germanic precursors?

Although it is grammatically redundant to use a possessive pronoun, since 'of' indicates possession, the phrase functions as an idiom. Using 'friend of mine' instead of 'my friend' implies colloquial endearment, affection, or closeness:

I'd like you to meet Tom; he's a good friend of mine.

is slightly more endearing and informal than

I'd like you to meet my good friend Tom.

of peter is a prepositional phrase, and therefore can function as its own adjective, modifying friend.

When you risk ambiguity, you might be better off saying Peter's friend.

The purpose of possessive pronouns and possessive determiners are to preserve the clarity. In the hyperlinked example, there is ambiguity: Mechanically speaking, the narrator is "not inclined to support Newt Gingrich's having served" but I doubt this is what the author meant.

So when deciding whether to use an apostrophe, just remember how it is affecting the word's part of speech, and therefore available uses. Friend of America has the prepositional phrase acting as an adjective. Friend of Peter's is a bit tacky and would be better off expressed as Peter's friend, in my opinion.

I tried to verify the premise, but at least from Google's corpus I couldn't confirm it with "friend of Peter." vs. "friend of Peter's." In the British corpus, there isn't a single instance of the former. In the American corpus both are represented, but "friend of Peter's" is still slightly more common.

The Saxon genitive was once a proper case in English. Nowadays it is a possessive clitic, as witnessed by the fact that you can add it to an entire phrase at once rather than inflecting each constitutent separately. As a possessive clitic, it doesn't really make sense in this idiom, so it's natural for it to disappear. Apparently this process has started at least in the US. If it has also started in the UK, then this hasn't reached edited books yet.

Personal pronouns are much more conservative, as witnessed by the fact that English still distinguishes three cases in its personal pronouns long after they have disappeared everywhere else. This is because phrases such as friend of mine are so extremely common that they are barely subject to regularisation. (It happens relatively quickly that a new generation has different grammatical theories. But specific common phrases don't change as quickly. We still use almost obsolete grammar in common phrases such as "little did I know", "far be it from me" and even "know thyself".) This may actually happen at some point in the future, but I would guess that a complete disappearance of "friend of Peter's" is required as a precondition for this event. This will probably take a while.

I can say that double possessive is neither idiomatic nor conscious / unconscious personal choice. This might be the answer to your question "Why we often add 's in forms like friend of Peter's".

In terms of syntax, the "s" in "friend of Peter's" is named "predicate possessive" and helps semantic interpretation with lexical information. Let's compare:

John's brother 

A brother of John's

From "John's brother" we can only infer "genitive relation" or "possession", whereas "A brother of John's" will give us more information e.g John has some more brothers.

But sometimes this type of construction might be elliptical.

That country was once Mary’s

may allow a reading equivalent to "That country
was once Mary’s country"" e.g citizenship, homeland relation – that was her home country until she emigrated.

For more details:

Partee, Barbara H., and Borschev, Vladimir. 2003. Genitives, relational nouns, and argument modifier ambiguity. In Modifying Adjuncts, eds. E. Lang, C. Maienborn and C. Fabricius-Hansen, 67-112. Berlin: Mouton de Gruyter.

Partee, Barbara H., and Borschev, Vladimir. 2001. Some puzzles of predicate possessives. In
Perspectives on Semantics, Pragmatics and Discourse. A Festschrift for Ferenc Kiefer