I believe both are acceptable: “I’m a big fan of Michelle Obama; I’m a big fan of Michele Obama’s.” But I favor the former.
Here is some guidance on the topic of double possessives from the AP style guide:
“Double possessives” must meet two conditions to be an acceptable grammatical construct: The word “of” must relate to an animate object; and the word that precedes “of” must involve only a portion of the animate object’s possessions.
The concept is probably best illustrated through examples: He is a classmate of Maria’s (correct because “Maria” is animate and “a classmate” is only one of several). There is also nothing wrong with writing, “He is Maria’s classmate.” And we can discern from the rules that it would be incorrect to write, “He is the only classmate of Maria’s.”
Other examples:
The brothers of Jane Smith preceded her in death. (All preceded her, so the double possessive is NOT appropriate.)
He is a benefactor of the settlement. (“Settlement” is not animate, so the double possessive is NOT appropriate.)
He is a brother of Jane’s. (“Jane” is animate, and “a brother” implies there are several, so the double possessive is correct).
While these examples might cloud the issue for you, in reality, we use double possessives frequently with possessive pronouns: He is a friend of mine. (Jeruba pointed this out.)
And if the employer for whom you’re writing uses another style authority, pay attention to what that authority says ;-)