Lissa, or Lissie, or
Lis; or
Mimi.
Mia could be made to work if you really loved it, but not unless!
How often do you 'refer to yourself' by a nickname anyway? Seems an odd thing to do. If you subtly bring it in, I very much doubt whether anyone would even notice.
Your best bet would be to tell your friends and family that you no longer feel like a
Mel so would they please try to get used to calling you
Melissa.