I have always liked
Becca.
Anyway.
Red is ok, but would make more sense if the child had red hair or something. I would never related it to
Rebecca.
Rex is a masculine name to me, so I find it hard to picture it as a nickname for a girl. It is also a stretch, and you would be explaining
Rex, Bex all the time. It would be easier to say you call her
Rex because she was born with small arms or liked to roar.
On that note though, I do like Bex a lot. I think Becks is believable and Bex is fairly obvious as well.
Others might be Reb, Rek, BB (my mother is
Rebecca and goes by BB sometimes). . . Sorry I can't think of more.