ARAM Summoner Spells, Runes & Masteries
After everyone is locked into their champion (i.e. the reroll period has ended), you should still be able to choose your runes, masteries and summoner spells for a while. This would make it slightly more strategic, in the sense that you would be able to optimize vs. whatever you're up against.
Right now, certain spells like cleanse cannot reasonably be taken because they could very easily end up being useless. Something like this would correct for that, allowing people to make more intelligent decisions, as a big part of ARAM is figuring out what your team needs in order to fight theirs.