It gets its name because you can use it to "beat the daylights" out of a target. This weapon is derived from a one-handed baton called a "Nightstick" by contemporary police.
The origin was that if you were hit hard enough by one you were knocked unconscious and it was "good night" for you. It was also called a "Billy club" because "The Bill" (slang for The Police) carried them as their issue weapon. This is similar to the name Bull Club as well. Because "The Bull" was slang for a Railroad Policeman,and these Bulls would carry clubs around to beat up law breakers as well. of these two Billy Club is the more common Slang term for a Daystick than saying Bull Club because police batons have not been called Bull Clubs since the 1940s.