Rivalry Events
There should be some more rivalries that boil over into the battlefield other than just the cat and bug and the cop/robber threesome, It would make the game feel more alive. Not to mention there's tons of people it would work with, for example:
Brand - Lissandra Ashe - Sejuani Shen - Zed Garen - Katarina Rumble - Ziggs/Heimerdinger Malzahar - Kassadin Thresh - Lucian Aatrox - Tryndamere Leona - Diana (cause... reasons.) Kindred - Kindred Teemo - All and a LOT more.
I think this would be a nice little thing that could be added, even if it's really just a kill counter like the vi/cait/jinx thing, though it could be something more substantial for people's who's lore is completely centered around revenge (Tryndamere to Aatrox). I mean it would add a little something else you can strive for in the game itself. I know if I'm on Rengar against Kha'zix, all I can think all game is "Imma get that fuckin' bug" and it makes it a bit more fun.