I am sure there is an idea or two in here that is already on the forums.
1. Including pre-game and post-game chats. There are plenty of "toxic" players who decide to wait until after the game to do their ranting/raving because they know that the chat is not shown in the Tribunal. If you feel that this needs more reasoning to get Riot staff working on a solution then feel free to add something but really this is all that should be needed.
2. Just remove the chatting feature completely. Obviously adding in a swear filter and the ability to block "ALL" chat hasn't done much. People still get reported for swearing by people who are too lazy to enable their filter or who choose not to so they can report people. People will find any way they can to report people and the chat box gives them many opportunities.
"He doesn't speak my language.'
"He offended me by using a word I don't like"(Usually involving swearing)
"He was being mean instead of giving me constructive criticism"(Usually used in games where people tried to help a failing player and that player didn't agree with what was said)
Obviously the only way to fully crack down in the way you guys want is to completely remove chatting. If people can't talk to each other then nobodys feeling can be hurt.
3. Find a way to group reports by queuing. We don't need to see multiple reports from friends defending each other when they shouldn't be but it is a common thing in the tribunal. If people are queued up together we don't need all of their reports ganging up on 1 player. Number 4 will have an example to show both this and the 4th idea.
4. Have the chat logs begin recording pings. Ping spamming is another issue that needs to be dealt with and it can't be seen in the chat logs. (And no, the current ping spam fixes do not work. Making someone unable to fire off 100 pings at a time by added a cooldown doesn't stop them from pinging as much as they can")
Player 1 and Player 2 are queued up together.
Player 1 ping spams player 3
Player 3: Stop f**king doing that.(Yes I realize most people would report and punish for swearing at this point but it is just for the sake of the example.)
Player 1: I am reporting you for swearing
Player 2: I am also reporting you for swearing.
In this example player 1 ping spammed which can be just as much of a game disturbance as anything else but he knows that the chat logs will not show that in the tribunal. He also knows that his friend(Player 2) will go with him on any reports he wants done. Having the chat logs in the tribunal showing these pings would be a step in the direction of filtering out toxic players.
5. Extending the timer required before skipping/punishing/pardoning Tribunal cases. 20 seconds is not enough time for a player to read the chat logs on multiple reported games. Having such a short timer just leads to people logging in and waiting a few seconds then hitting punish. Punish enough games and eventually you will be with the majority to earn yourself some extra IP. There will probably be people that come into this post and claim that this does not happen. Try looking through some of the cases posted here. There are actually some innocent ones who are punished and it is most likely caused by this happening or something similar.
6. Just remove the "Unskilled player" report option. Surely by now everyone has figured out that this is simply a troll report and is not taken seriously. Now it just takes up space in the drag down menu.
7. Require people to actually leave a message/note in the reports. Blank reports are not very helpful, it would be very helpful for the people doing the tribunal to see a report and know the reasoning for it. If you are going to report someone for offensive language point out what they said. Want to report someone for griefing? Say what they did. This helps people who try to judge tribunal cases fairly.
I am sure people will pick away at this list. Pointing out flaws or reasons why they don't think they would be good ideas. There will also be downvoting and possible upvoting. That is nice I just ask if you are going to do something like that please post your message explaining your reasoning. I am aware that there may be problems with my ideas so adding any counter ideas would also be helpful. If we keep this post clear of spamming/pointless messages maybe the Riot staff will actually read it.