I think we just need to swap ARAM to ADAM and it will all be fixed. That, or change make changes such that you get a pool of maybe 30 champs that the game will choose from for your "random" champ.
Ok, so, this is what the problem is: it's feeling irrelevant. It's when you get udyr, or poppy, or really most melees and the enemy team with their nidalee and and morgana and varus just punish you every single time for trying to play. That's the problem. Not the game mode, the game mode is lots of fun: constant teamfights are awesome, and it sure does make you pick up your game in terms of skillshot dodging. It's the fact that by random luck, you can be so totally outclassed by either your enemies OR your teammates that's the problem. There's any number of fixes, but the reroll system is absolutely terrible, as is the random aspect of it.
Note about the reroll system: it severely punishes you for owning many champs. So, i've got 84 champs, and I want to play/will feel relevant with less than a third of that number (those are the ones i'm good at or are really strong on that map). Tt takes me 3 games to gain a reroll. Someone with only 28 champs (1/3 of mine) could play a champ that's relevant to this map almost every single game, and still get a reroll as frequently as i do. What's more, since they get the champs they want more frequently, they use less rerolls, and therefore, even when they do get a champ they would rather not have, almost always have the resources to reroll to a champ they DO want to play, where as I get stuck with a much lower chance of feeling like i'm making a difference, and the same games to reroll ratio (or nearly, if i use both i'll be one or two games ahead.. but they wouldn't have needed those rerolls in the first place).