So, we all love and hate this mode because as fun as it is, it's a great way to put a really sour taste in your mouth playing a casual 'fun' mode but getting completely steamrolled because your team dice gods say you get 0 carries or 0 cc champions, while the enemy team gets 3 marksmen or literally a stun on every player.
So, my SUGGESTION (Read: Do this if you want to make this mode better with literally no complaints from anyone) is to have the ARAM algorithm have a table of values for carries/supports/etc that breakdown their power levels in terms of damage, tank, cc, etc.
Use that, and put in a spread of 1/2/2 or 2/3 so that a matrix table fills out a roughly even playing field for both teams, and then randomize the players.
Also: Hard ban out champions like Ezreal, Kai'Sa, Teemo; Champions that when given massive cooldown reduction and no mana costs have way too much power just on the merit of their basic kits. Teemo is a great example, because if you even hope to fight him you need a boatload of MR, for one, and two, you need to NOT be a champion that deals damage with autoattacks (His blind will always outlast the cooldown.)