People say both, although the correct form is (oddly enough) "he's a friend of my brother's." This is more obvious when we use a pronoun "he's a friend of mine." No native speaker would ever say "a friend of *me."
2017년 9월 16일
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He's a friend of my brother.
He's my brother's friend.
He's a friend of my brother's (what?). --> this sounds like an unfinished sentence to me.