The basic answer is this: in "Rules of the Game," Waverly is attracted to the chess set because of its unique rules and how she relates these unique rules to her role as a first-generation American in the Chinatown community in San Francisco.
In order to more fully understand my answer, let me take a step back and discuss chess.
As a general rule in literature and film, chess has symbolic value. Each piece of the game has a specific purpose. The pawns are sent out in front to be taken, while the king and queen sit in the back and kind of do whatever they want. In addition, chess is not like checkers in that each side is always on the offensive. In chess, defense and strategy are key. So whenever you see chess in a movie or read about it in a book, expect a certain amount of scheming to come from the characters playing the game.
In the Chinese immigrant community Waverly grows up in, she is clearly attracted to chess because of the rules placed on her growing up. In fact, when the kids first open their used chess set, Waverly becomes a bit obsessed with pawns and asks why these pieces can't do certain things on the chess board, which causes her brother to say, "This is a game. These are the rules. I didn't make them up. See. Here in the book." Waverly's connection to the pawns might also help readers understand why she becomes so upset with her mother later in the story.
But once Waverly learns the rules in chess, she is able to manipulate those rules to become a prodigy. However, Waverly's mother is just as good at manipulating the rules of the game, but not chess. Waverly does make a direct connection between her mother's scheming and chess:
"Her black men advanced across the plane, slowly marching to each successive level as a single unit. My white pieces screamed as they scurried and fell off the board one by one."
Being the child of an immigrant is not easy. Waverly has to learn two sets of rules—her mother's and America's. Her connection to the game of chess symbolizes this tension.