Some things are difficult to practice, some things are boring to practice, so these are suggestions for mini games that I believe would improve the experience of practicing two areas that are crucial for success in League of Legends:
1. Draft 5v5 Teamfight/Teamcomp practice on the howling abyss. This would be good because you would go in to this with the mindset of practicing positioning in team fights, picks and bans, and the limitations of your team comp verses another team comp.
2. Last Hit practice. The game could be made to look like a tetris style arcade game, where minions at varying levels of hp fall towards your champion and you must only hit the ones that need to be last hit while dodging the other minions. This would be good because it would teach people how to recognize when to last hit and how to dodge skill shots, as well as last hitting under pressure.