Special Game Modes (e.g. URF) and the free champion rotation
I think the free rotation should rotate faster during a custom game mode. There are often a lot of champions players would love to try on a custom game mode that they don't own. It isn't practical (or even possible) for relatively new players to buy any / all of them.
The free rotation works great for normal play as it gives you an extended pool of options for a week period to practice with a champion and see if it's worth buying. When it comes to custom game modes though, if champion X isn't in one of the 2 weeks of rotations out during the mode you don't get to play them at all if you don't own them.
With how short these game modes last, I think it would make sense to accelerate the free rotation changes during the mode to give players access to a larger variety of champions. People who own more champions still benefit from having access to those champions for the entire duration of the game mode, while newer players get to try a large variety of champions to ensure access to highly desired picks for a couple days while not giving them access to a specific pick for the duration if they don't own it.
E.g. with the current URF: Highly popular picks might include: Shaco, Wukong, Evelyn, Sona, Blitzcrank, Hecarim, Vladimir, Ezreal, Nidalee, LeBlanc. Rather than Hecarim being available for the entire first week and none of the others ever being available if not owned, each of the champions would be available for a short duration meaning people who own, say, Shaco still get 90% of the time being able to use him while no one else can, while people without a large champion pool get short access to each of these champions.
I think most players just really want to try a few specific champions once or twice in a custom game mode rather than spamming games with the same champion, so overall this would increase enjoyment of the game mode. It may even solve the problem Riot notes of interest in game modes dying off over their lifespan - players may get bored with their champion options for the mode rather than the mode itself.