If we change the phrase from "a friend of Molly's" to "the friend of Molly's", then I think the additional possessive does perform a function. Compare:

That's the friend of Molly who always falls down the stairs at parties.

That's the friend of Molly's who always falls down the stairs of at parties.

I think the second sentence leaves us in no doubt that it is the friend who gets legless rather than that fine upstanding specimen of womanhood, Molly, while the first sentence causes some momentary doubt.

Also I think there is an implication in "the friend of Molly" that this is her one and only friend, while "the friend of Molly's" lets us know that Molly has lots of friends, of whom this is one.

Bingley

Bingley