"The Dead-Eye Kid"
Without telling his father, Mark helps a boy from Brooklyn hide from the law when the boy is accused of murder.
From "The Rifleman Episode Guide List"
By the end of the episode, it looked like Don Mahoney was going to stay out West and make a new life for himself, rather than go back to the Bowery in Manhattan. However, he may have made trips back home on occasion to visit family, like to a brother, perhaps.
If Don did have a brother, he would probably have started his own family and his family tree might have eventually led to a grandson or great-grandson who fell into the life of organized crime as a 'Wiseguy'. There is such a mobster by the name of Mahoney in Toobworld, who once made a huge bet that he could keep a dumpster cat inside his pants. That cat turned out to be more feral than he expected, which is why he came to be known as "No Money" Mahoney".
Don Mahoney could have been the source of No Money's family tree, as well. As young as he was when his path crossed the McCains' (probably at best nineteen?), he may have frequented a working girl back in the Bowery - or just a rather loose-living classmate - and left her with child.
By the way, the producers of the show missed a great opportunity in naming this episode. "The Dead-Eye Kid" doesn't give a clue to its content; but "The Dead End Kid" or "The Bowery Boy" would have been a great play on those movie titles.