The abuse that Bruce suffers at Brian's hands as a child is a major contributing factor to his eventual transformation into the Hulk. When Brian was a child, he and his sister Elaine were bullied and abused by their father. Brian did not want a son, confident that he will have a mutant son. When Brian was little, he was abused by his father, Bruce Banner I. Because of this, he bullied his son, Bruce Banner, named after his grandfather.
In the 2003 movie, Hulk, Bruce Banner's father is named "David" Banner. The character is actually based on Brian Banner, but was changed to "David" to pay homage to the TV character (although sort of backhandedly).