There has been a lot of discussion about ARAM smurfs. Riot tried to fix with this by giving people with more champs a re-pick option more often, but this really still doesn't discourage the creation of such accounts.
My solution to this problem is, that every champ should be available in this mode. It wouldn't affect other game modes in any way, it would encourage the use of own accounts because players most likely have most of their runes on that account and it would force the play of underused champions in a way. And after all, the mode is called ALL RANDOM all mid :)