I've read a bit about the reason for using the double possessive but there are still a few aspects that don't work for me:
If the difference between "Joe is a friend of Jim's" and "Joe is a friend of Jim" is the active person (with Jim being active in the first sentence and Joe in the second) then the sentence "Joe is a friend of Jim's" should have the same meaning as "Joe is Jim's friend," and the sentence "Joe is a friend of Jim" should have the same meaning as "Jim is Joe's friend." Using the same logic, the sentence "Joe is my friend" should have the same meaning as "Joe is a friend of mine" and the sentence "I am Joe's friend" should have the same meaning as "Joe is a friend of me," which we would never say. Furthermore, one would never say "Jesus is the son of the Lord's" when they mean "Jesus is the Lord's son;" rather one would say, "Jesus is the son of the Lord."
If the difference between the two is that "Joe is a friend of Jim's" implies that Jim has multiple friends while "Joe is a friend of Jim" does not, then the same issue arises when I try to say "Joe is a friend of mine" without implying that I have multiple friends I say "Joe is a friend of me" which, again, one would never say. Also, you wouldn't say "Jim is an owner of the club's" when you mean "Jim is one of the club's owners."
If two conditions must apply, that is the double possessive can be used only if the word after of refers to an animate object and the word before of involves only a portion of the animate object's possessions, I still can't say "Joe is a friend of me" if Joe is my only friend or "Jim is a friend of him" if Jim is his only friend. Also, can't one say "Joe is the only friend of Jim's?"
Also, I don't want to hear that the second apostrophe is the only way to distinguish between "a picture of the king" and "a picture of the king's." In the sentence "a picture of the king" the of is not a possessive; you can't say "the picture of the king's."