I'd love to see a more 'normal' boy like Tony Calderone (from the NES) in the role of Protagonist....A kid with a basically normal upbringing (Mother + Father or maybe Widowed Mother), a normal background (a non-trouble maker), average or better looks and height, a more normal disposition, a lot more smarts....An average kid who is thrust into the role of 'Headboy' and must deal with all the craziness of Bullworth Academy, who must really work to keep the peace.
Jimmy Hopkins is portrayed from the start as a serial troublemaker and a not-very-bright Anti-Hero, who for some unknown reason does missions and errands to help others. While it is true he is compensated with a monitory 'Reward', it still really goes against character that he should be helping out anyone. As demonstrated in the game, there are other ways to make cash (Lawn Mowing, Delivering Papers). The problem of money could be muted by having an allowance made available at the start of each chapter.
While many of us consider ourselves 'normal', most of us have plenty of stories of the 'weird' and 'crazy' kids we knew in High School. While there are High Schools in the U.S. just as violent and dysfunctional as Bullworth Academy (mostly in the past and in heavily populated areas), most schools are just mild hellholes at best. A game with a little less fighting and a bit more problem-solving, a game with a more coherent storyline, would in my opinion be a much better game.
Don't get me wrong, I loved Bully....But dealing with the unknown and having intrigue and suspense just would make it a whole lot better.