Becca is a shortend form of the given name Rebecca. The name is of Hebrew (i.e. Aramaic) origin, meaning captivating or to shine. The original Hebrew form is usually transliterated as Rebekah. Rebekah was the wife of Isaac and the mother of Esau and Jacob in the Old Testament.
Becca is not used as a diminutive form as often as is Becky, another shortend form of Rebecca. Instead it is affected by a number of Rebeccas as the preferred form of the name. (My sister is an example: she objects to being called either Rebecca or Becky, though her given name is Rebecca.) The use of Becca seems to be a fairly uncommon, for Becca's GCUI (vs. Rebecca) as of May, 2001 was 125,000/2,170,000, or 5.8%.