And by quests, I don't mean the likes of Bilgewater quests (which was nice, don't get me wrong). It's more like challenge, I guess?
I mean things like "clear the entire jungle in 1 minute 30 seconds as Nidalee," which promotes junglers experimenting with different setups, and getting a good statistics on how fast certain champions can clear jungle, as well as finding out optimal rune/mastery setup. You can also check exactly how low elo players are underperforming compared to high elo players in champions that require some decent amount of mechanics and optimizations, like Nidalee and Rek'Sai.
Or maybe "do 1500 damage to the dummy (changes for every quest) in 2 seconds with 5000 gold as Lissandra," to get people to try out underplayed champions (I don't play Lissandra and I don't even know if doing that much damage is possible, or too easy; pls no flamerino).
Riot can use these quests to figure out what's wrong with the game. It could give hextech rewards, but if you don't want to give out free stuff too much you could just give out some free icons, or maybe even champion mastery points based on highest performance. Or just show leaderboards so people can compete even without any rewards.
Seems like a win-win for both players and Riot. Could you please implement this?