For sure giving it an online direction would be good but only if it didn't come at the cost of the single player experience.
In the case of single player there's a lot of things I'd want but to keep the list short...
though its still fairly long if you look at each feature.1. Being able to more greatly customise the characterPerhaps even allowing a variety of heroes or making it so that the heroes personality changes based on the apporach you take. Included in this is being able to change the characters fighting style and having to make choices to give the character strengths and weaknesses or making them an all rounder, both physically and academically (and perhaps balancing this out), definately a must for an online experience in any case. Obviously people (and thus characters) have classes they like and dislike, and classes they are good at or not... so why should this be any different?
2. Being able to join cliquesAt the very least though you should be able to change your standing with various non mission related activities. During the story the hero should be presented with a variety of ways to solve a problem, either through fists or through diplomacy. It annoyed me in chapter 3 that Jimmy was playing nice with the Greasers then listened to Derby and Lola. An alternative would be playing nice to get what you wanted from them. Overdepending on either violence or diplomacy might give difficulties down the road however

. Too violent and you get a bad reputation and few people will trust you, be too diplomatic and people will use and betray you. Not sure how best to develop this but... just putting it out there.
Regardless, if you were to join a clique, obviously getting that clique into a position of power and/or you in control of the clique would be the goal. This would present different ways the story could end for each of the cliques 'ruling/pacifying' the school, or the lone hero/few others actually pacifying the problem.
3. More cliques really means another schoolI read another thread of bully ideas that added more cliques and made alliances and enemies with these newer cliques. Really I don't mind it staying at Bullworth (some people do mind) as long as we get a new hero (nothing against Jimmy but keeping it with him forbids customisation and I think it ended well with him) but to suddenly add more cliques it really requires a new school and it would make the expansion of cliques make sense as well as the alliances that were set out. Some people would gel, others would not

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For online... I think a more cooperative experience would be good. You could add a sense of competition to it as well for various things but cooperative multiplayer would be a lot of fun...
Bully 1 was much too short for sure as packed as it is with content and character. Extending on that line of design for both more of the same and added multiplayer would be surely the way to go yes?