Even though the brothers were named as a joke by their father, their names prove to be the central image in the play. The names are obviously connected to history and establish the roles of each brother. Just as John Wilkes Booth assassinated Abraham Lincoln, we know Booth will kill his brother Lincoln. The tension is then established in the play, and we intensely watch and wonder what causes Booth to kill his brother.
Booth, the underdog, wants to be a hustler of three-card monte, and he constantly tries to prove he will be a better hustler than Lincoln if Lincoln will just show him how. This is why Booth invents his name of "3-Card". He has no sense of identity, of who he is. By allowing no one to call him anything but "3-Card", he boosts his confidence to be better than his brother.
Assuming his new identity of "3-Card" is the only way that Booth is able to kill his brother in the end.
Lincoln, the topdog, is satisfied with his job at the arcade. He was a former master of three card monte but gave up his hustling when his stick man was killed. He understands that in three-card monte, as in life, there is no winning. Booth never learns this, being overly confident and eager to prove himself the equal of his brother. When he sees Lincoln has hustled him, Booth can't take it and kills his brother.