I have been looking back over this discussion, and there is one possible explanation that no-one has mentioned.

"A friend of my sister's" implies that my sister has two or more friends and we are referring to one of them. So we mean "a friend of [= from among] my sister's friends".

"Of" here means not "belonging to my sister's friends" but "part of the group of her friends". This is why we cannot say "the friend of my sister's" – there cannot be a group if she has only one friend.

So I think the double genitive is perfectly logical.