Sports are a point of contention in Gene and Finny's relationship. Gene continually describes Finny's incredible physical prowess, in addition to naming various trophies Finny has won for his athletic skill. Gene, on the other hand, mortifies Finny by only achieving the post of assistant manager for the rowing team. Gene states that he doesn't really enjoy sports, and often only plays for Finny's sake. Gene also feels (at least at one point-he's not a reliable narrator) that Finny is out to destroy him, and sees sports as Finny's way to distract Gene from his studies, causing his grades to slip. This is actually the motivation for Gene pushing Finny from the tree.
Finny's invention of blitzball highlights the dynamics of his relationship with Gene, as well as Finny's own innocent nature. He creates a game simply for the love of sport, changes the rules when it suits him, and seems to be genuinely content watching others enjoy themselves as well. He is the altruistic character in the novel, giving and never asking in return. Gene, on the other hand, is thrown off-guard by the game of blitzball. He is confused by the constant changing of the rules, and ends with a feeling of frustration, which is later directed against Finny himself. This is how their relationship develops as well.
One last note-the name of the game underscores the reality of WWII for the boys. Finny scorns the war before his accident, but later tries to join the military, almost seeking validation of his usefulness. Here though, the name "blitzball" suggests that Finny still sees the war as unreal, something beyond the walls of Devon, something that cannot touch the students. Thus, he makes a game of it.