Sometimes editors (and others) wonder what the difference is between, say, “He’s not a fan of Cher” and “He’s not a fan of Cher’s.” Is there a distinction? Is it equally important in all instances?
There is a distinction: it’s between possession and association. In some cases it’s the same thing; in others, quite different. “A picture of Mr. Goldfine” is not a picture belonging to Mr. Goldfine but a picture depicting him; “A picture of Mr. Goldfine’s” is a picture belonging to him. (“Mr. Goldfine’s picture” can mean either because we use the “possessive” for both possession and association.)
When you talk about fandom, there is again the possible distinction between association and possession, but in that case it really refers to the same thing, just from a slightly different angle. “A fan of Cher’s” is the same as “a fan of Cher” but in the “Cher’s” case it gives a sense of there being a collection of fans belong to Cher, as opposed to it being simply an attitude on the part of the fan.
It also follows that because running in the rain is a kind of action, not an entity that can possess, “A fan of running in the rain’s” is odd.
English pronouns are more archaic than the rest of English; they preserve case distinctions that have been lost everywhere else, mainly because they’re so entrenched and we used them automatically by habit and without analysis. In cases such as this, a distinction can be made with them when there is a real distinction to be made: “A picture of him”; “A picture of his.” In instances where the distinction is not a significant one, we may hew to the older construction, which in this case uses the genitive because that was the case governed by this construction: “A fan of his” may seem more natural than “A fan of him” (though this will vary from speaker to speaker). (Languages that have full and productive cases systems for nouns tend to use different cases after different prepositions and depending on context; German and Latin are two languages that do this. Old English was another.) Note, however, that the association/possession distinction still matters: “I am not a fan of it” is fine; “I am not a fan of its” is probably not.