Arthur, William (1857) An Etymological Dictionary of Family and Christian Names. New York: Sheldon, Blakeman. Public Domain.
We don't have any stats on how common this name is. This is probably because it's very rare in the UK.
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Related and similar surnamesBabcoc
The Babcock surname in historical dictionaries
An Etymological Dictionary of Family and Christian Names (1857)
Little Bab, or Bartholomew; from Bab, a nickname for Bartholomew, and cock, small, little, a son; cic, cock, el, and et are diminutives, and include the ideas of kindness and tenderness, associated with smallness of size. It may be from Bob, the nickname for Robert; Bobcock, the son of Robert, Robertson.