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 surnamesConan
The Conant surname in historical dictionaries
An Etymological Dictionary of Family and Christian Names (1857)
(origin: Welsh and Gaelic.) Conan, a river. Counant, a cataract in North Wales, from cau, a chasm, a deep hollow, shut up, and nant, a rivulet.