To be honest, they all sound natural. It simply depends on who you are speaking to as to which one to use. If the other person knows your friend, maybe use "my friend." Otherwise, use "a friend of mine" or "a friend." This is similar to the use of あの and その in Japanese.
You'll most likely use "I'm meeting a friend tommorow" if the person you're speaking to doesn't know the friend you're talkimg about. You'll also most likely use "I'm meeting with my friend tommorow" if the listener knows the said friend but isn't friends with the person. The last exampke sounds really formal to me, it seems like the kind of thing I'll write in a formal letter. but then again, it's not set in stone, anyone is usable in conversation, but "...a friend of mine" seems the least casual of the three.
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