It's a classic, solid name -- as long as Becky isn't used. Becky is becoming a joke name this decade.I much prefer the spelling Rebekah. Reb is a cute nickname option; I've known Rebs and it wore nicely on them.
Likely just with younger generations, but look up "Becky with the good hair". This use of the name has led towards 'Becky' being used as a pejorative. It's especially common on the internet.