Another idea for down-sizing ARAM accounts
Instead of making all the champions free for ARAM(the popular suggestion), and instead of giving everyone a random champion pool page(my own suggestion), how about...
...free to play champs for the week cannot be rerolled, and they cannot be traded. I've noticed that the ARAM accounts players prefer to save rerolls for when they get free to play champs that they think are bad, it's pretty consistent. If they don't have rerolls and they get a free to play champ that they don't like and you have one of the popular ARAM champs they will definitely try to trade for your champ.
So, a good way to give them all a kick in the butt every so often is to just deny them being able to trade or spend their reroll when they get a free champ. I mean, it happens more often to players that own fewer champs, therefore it happens more often to ARAM account players on average.