I think friend of mine can be translated to my friend. In that case, doesn't friend of me make more sense?
If we translate friend of mine to one of my friends then I guess friend of mine makes sense for my friends being mine.
Is there a difference?
When do you say ... of mine instead of my ...?
Is there a specific situation when you use one or the other?
In friend of mine, mine means my friends, so literally He is [one] friend of [all my friends]. or, more idiomatically, He is one of my friends.
If I only have one friend and he is my only friend, we cannot then say "he is a friend of mine", because the "mine" doesn't mean a group of people. If I introduce my daughter I would never say This is a daughter of mine. but This is my daughter.
Mine is used to refer a thing belonging to (or a person associated with) the speaker.
Since it is about a friend (who is associated) mine must be used instead of me.
Adapted from NOAD
Another usage:
Is there a specific situation when you use one or the other?
"Friend of mine" would generally be used when you're saying something like "Kim is a friend of mine," in other words, at the end of a phrase or sentence. "My friend" is often used when saying "Kim is my friend", or in the construction, "My friend, you have a rip in the back of your shirt."
Ok, sorry, long two-part idea.
"of someone" in this case is not really about possession. My impression is that English speakers do not like to use "of" simply to show possession; we have other grammatical constructions for that. Instead, "of" is used to refer to patterns of association or constituency in some larger whole, as in "out of" "consisting of".
"a friend of mine" is really a shortened form of "a friend of mine (friends)" as argued by other comments. This means, literally, "one friend out of my several friends". The OED etymology sections have a long but clear explanation of the history of "mine" and "my" that more or less clears this up. Historically, "my" originates from a singular version (min) and "mine" from a plural version (mine) which were otherwise grammatically the same. Therefore, "mine" probably does have something to do with the plurality of friends in this case. Just like "mine eyes" probably made sense in the the past, but not "mine nose". One thing that either clinches or messes up this interpretation is that people say "of hers". OED calls this a "double possessive" which developed by extrapolation from 's. But some dialectical variants use "of hern" instead, which was definitely plural in the past. Therefore. "hern friends" was used in the past. Was "hers friends" ever grammatical? I'm inclined to think this is (historically) about plurality and not about double possession. Maybe phrases like "of John's" were originally in the plural possessive sense "of Johns" where John has many of something, not where there are many Johns, but were reanalyzed as a new special kind of possessive when this kind of plural possessive was lost. Therefore, the idea of double possession is a later way of explaining why people speak like this.
"Mine" was once used to mean "my." ("Mine eyes have seen the glory...") I would guess that this idiom is a holdover of that usage, although here it functions as an object rather than an adjective.
Why "of mine" rather than "of me"?
Evidently, it has to do with the function of the preposition "of" in standard modern English.
"That was kind of you to say." Here "of" creates a relationship between "you" and "kind", but does not indicate possession. It simply answers the question: "of whom?".
On the other hand: "Those were kind words of yours." Here, "of" indicates possession, inasmuch as it answers the question: "whose?"
There are certainly English dialect forms that conflate "of whom" and "whose", perhaps in the same way that informal modern German frequently uses the dative instead of the genitive case.
Do you mind me/my closing the window? This illustrates a case in point where standard modern English acknowledges the similarity of both forms.
The nominative form of "me" is "I". You wouldn't say, for instance,
"He is a colleague of me."
Any more than you would say
"He is I colleague."
You would say:
"He is a colleague of mine"
and
"He is my colleague"
The same goes for friends. For whatever reason, friends, colleagues, etc., are treated as possessions in English.
As noted above, when it's third person, you can do this. But consider:
"A friend of Jane."
and
"Jane friend."
Again, it doesn't work.
Perhaps this use of the non-genitive is just a matter of usage. I'm not sure.
X of me is always ungrammatical, and I don't think there are any English constructions that use of me. The emphatic possessive construction is formed by of mine, and the equivalent using other persons (of yours, of hers, of Jane's, of the cat's, etc.)
There isn't a logical rule that explains why this is, and there doesn't need to be. That's just how the language works.
