In English phrases like
Jesse is a friend of mine/*of me
the case of the word "mine" is not the oblique ("me") which usually occurs with prepositions ("That's a part of me that you don't see too often")
What licenses the genitive case ("my/mine") in possessional phrases with the preposition "of" instead of the oblique case ("me"), which most prepositional phrases take?
A few more examples (where the # ones sound a bit odd to me, but that just might be semantic satiation messing with my judgements):
I am a friend of Jesse's/#of Jesse
I like that girlfriend of Jesse's/#of Jesse
I'm a friend of the king/#of the king's
She has the voice of an angel/*of an angel's