1: Need Pre-game chats, people typically trashtalk or act the ass in pre-game chats, this would help for a better judgement
2: Post game chat, pleaseee, I constantly see reports of people talking about people acting out in post game chat, even if they acted perfectly nice in-game, unfortunately, we can't figure out if they are telling the truth or not, without being given the information!
3: Inappropriate name reports, like, really, I've sat there, staring at a case for 20 minutes, TRYING to find where they listed the summoners name, it's impossible to really see if the summoners name is inappropriate if you can't see the name, if your worried about players learning about other players, then simply make it to where the names are only seen during inappropriate name reports! Like really D:
4: Make the possibly most severe cases go up first, the one's that receive multiple reports, I've seen a few cases the first couple games out of 5 per-say, were mild, and not really ban worthy, but the 5th game alone just made me sit there and go WHY?! was this person pardoned? Like this one http://na.leagueoflegends.com/tribunal/case/5959171/ the first couple of games on this one was mild, but the last one completely bewildered me on why they got away with a pardon, even 1/5 games especially with how fairly bad the last game was, should have been a punish, any toxic game should be punished :|
There's probably more that could make things easier, but honestly, these are the main three I've run into.