Double possessives are a strange quirk of English, but useful nevertheless. If I write, "A picture of my brother", I'm probably referring to a likeness of my brother, but, if I write , "A picture of my brother's", I mean a picture that's owned by my brother. (And if I write, "My brother's picture", I could mean either of those.)
In your example I don't think it matters which style you use, as both make the meaning clear. On balance I'd probably use the first one, because it parallels "Friends of mine" (double possessive), rather than "Friends of me" (single possessive) which doesn't work.