Why isn't ARAM a 1v1 gamemode?
I know the lore surrounding the gamemode, where people were doing it in Summoner's Rift games, so they made a map for it, but honestly, ARAM would be the perfect gamemode for 1v1s... with a bit of tweaking.
Often when people 1v1 they pick their main champ. But if you get a random champ, it's just raw player skill. And it's 1 lane, so you don't need to worry about somebody picking up a free 850 gold by going to a side lane and free pushing for a tower.
You could even make it where people queue up for ARAM's (the proposed remake, of course) as Spectators, so that they could spectate the 1v1s in live action, and comment with other spectators in a chat box like twitch.tv has.