The Ballad of Lefty Brown

Lewis Pullman who plays Billy Kitchen is Bill Pullman's son in real life.

Although Lefty does not appear to be left-handed -- he shoots right handed, rides right-handed (holding the reins in his left hand so that his right hand is free), and wears his sidearm on the left side, butt forward, so that he can draw it with his right hand -- he could have gotten the nickname for some other characteristic or habit.