Clay itself is often a nickname for Clayton. If you name your kid Clay by itself, well, that's nickname-y enough.
Famous people named Clay:
Singers Clay Aiken, Clay Cook, and Clay Walker; football players Clay Matthews II (and III); baseball players Clay Buchholz and Clay Hensley; actors Clay Greenbush and Clay Clement.
The word (and name) Clay comes from the Germanic root "glei" which means "to stick together."
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