Atticus’ pocket watch is a family heirloom, symbolic of his southern heritage. He intends to pass the watch down to Jem when he is old enough. In this way, it represents the continuing of old traditions as well as Atticus' social position. It is significant that Atticus removes his watch and chain during the trial, so he seems less high and mighty to the rural people on the jury.
The watch could also represent the passing down of moral values from parent to child. Atticus is a role model for his two children, standing up to injustice and being his own man in a world where that can be a dangerous thing.
While the watch isn't one of the bigger symbols in the book, it is quite significant in the Finch family. Its significance can be found in the scene where Scout asks to play with her father's watch and he tells her about what he intends to do with it someday.