This dog is a friend of mine's.

Apparently the apostrophe (plus the "s") is modifying the whole phrase "a friend of mine" even though it looks as if it's modifying only "mine". (mine is already a possessive so it wouldn't make sense to add an apostrophe to it)

My question is "How do we call this use of the possessive?".

BTW, it seems that we cannot say "This dog is my friend's." so we're forced to use the above construction or something like "This dog belongs to a friend of mine". (objective case)
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Comments  (Page 5) 
AvangiHappy New Year! - A.

And to you! I'm well behind with all my work at present, but I haven't forgotten your questions. I need to do a little research on this topic and will get back to you asap.