The name "Arthur" does not occur in any context before the 7th century A.D. Once it starts occuring in historical records, we generally assume that these people are being named after King Arthur. This begs the obvious question of where King Arthur got his name from.
One posibility is that "Arthur" derived from the rare Roman name Artorius, meaning "plowman", but another intriguing possibility is that it derived from "Artgur", or "bear-man", Art being an old Celtic root meaning "bear" and gur a Brythonic word for "man". There is some circumstantial evidence for such a contention.
In numerous early Welsh works, art commonly serves as a figurative synonym for a warrior. Moreover, linguists attest to the softening of harsher sounds such as "tg" to the "th" sound in late Old English. Finally, a gloss added to a late 12th or early 13th century manuscript of the Historia Brittonum states that "Arthur" means ursus horribilis, or "horrible bear," in Latin. Thus, it is plausible to conceive renowned warrior (as Arthur is supposed to have been) receiving a nickname or nom de guerre such as Artgur, only to have the nickname conflated with his given name in the popular memory.