Based on these two examples:
"Sweet child of mine" "You wouldn't know him, he's an old friend of mine"
One can say the following about these uses of these stand-alone possessive pronouns.
of mine, of theirs, of ours, of yours, of his, of hers, [of its, very unusual].
They are used when the referent (the person who one is referring belongs to a class of persons). It is used to be emphatic, or, in contrast to someone else or something else.
A friend of mine = One of my friends [same thing]
A friend of mine has that CD. = One of my friends has that CD. Those two sentences mean exactly the same thing.
This is very different from:
My friend's CD was not in her living room.
My friend's CD= the CD that belongs to my friend.
"He's an old friend of mine" , therefore, can be expressed as: He's one of my old friends. But cannot be expressed as using a noun plus apostrophe s and a noun, as in: my friend's CD.
This is all very standard English.
A friend of his came to the party late. = One of his friends came to the party late.