prepositions - In “an idea of Hölderlin’s”, is “of” a partitive or a genitive?

We often use these constructions. A friend of mine is probably the most common.

I have often wondered, being an English teacher, whether the function of the preposition of in such contexts is that of a partitive (it implies an idea of splitting something into parts, so a friend of mine is a part of the whole of my friends) or a genitive (it implies the idea that something belongs to something else, so an idea of Hölderlin’s would mean Hölderlin’s idea).

I realize the distinction is subtle. Is there a professional scholar in linguistics here who can clarify?