What was his name?

I think it was Cousin Bessie.

Actually, there were TWO chimps featured on "The Beverly Hillbillies" during its run...the b&w episodes featured a boy chimp named "Skipper"...he was soon joined by a baby chimp named "Cousin Bessie", who was featured for the rest of the series's run. (I have no idea if they used the same chimpanzees or if they had a variety of "Skippers'' and "Bessies" hanging around the set. But these are the only two that I know of.)