"King of the Bingo Game" refers to the nameless protagonist of the story, who, as he continues pressing the button controlling the spinning bingo wheel, forgoes his African-American identity for a new identity tied to his ability to control the game. He reasons that "King of the Bingo Game" is a far better title than the white man's name given to his grandfather "a long lost time ago" (66). In a way, it's as though the protagonist is being re-born, this time with a sense of agency: he gets to give himself his new name.
But let's dig deeper. We can interpret "King of the Bingo Game" to refer to the protagonist's feelings of mastery over his situation. He is pressing the button. He is controlling the wheel. He is controlling his own destiny. But there may be some irony at play here. A bingo game isn't exactly an "illustrious" activity. Thus, on the one hand, being King of the Bingo Game might come across as tragic or pitiful.
But on the other hand, the author might be saying that being king of this everyday game, on his own terms, is better than anything his predecessors had – and that says a lot. While bingo might seem unimportant at first blush, the game really means something to our protagonist – and possibly all those whom he represents.