ARAM pool includes champions you don't own as well.
This is what I originally thought it was, and was disappointed to find out it wasn't. It would be more random, and be more about trying something new and finding the champions you like. It's not like buying specific champions does anything for you in ARAM anyway. And what percentage of League games are ARAM, or what percentage of people only play ARAM? It's not like it'd be a huge loss to Riot. In modes like Nemesis I'm pretty sure the champion pool wasn't limited, I played Urgot loads of times and I don't own him.