I like
Rebecca. It's a strong, stand up name.
I know quite a few of varying ages, so it doesn't seem dated to me. It seems that
Becca is more common a nickname among the younger Rebeccas. I once had three in my year level that all went by nicknames: Beck,
Becca and
Becky. You can avoid it all you want, it's up to the girl what she wants to be called. If she introduces herself as
Rebecca then people tend to get the idea.