While Luke Cage eventually makes the connection when Diamondback arrives on the scene later in the first season, the reason the Harlem crime lord is called Cottonmouth is because he got a few teeth knocked out when he was an up-and-comer.
Purposeful as the tongue flicking may seem, it is worth mentioning that Cornell “Cottonmouth” Stokes’ nickname doesn’t actually have anything to do with snakes.
Because of this origin story, Cornell actually hates the moniker (he probably would have embraced it if it had anything to do with being snake-like).