ARAM solution: require a minimum number of champions owned depending on account level
^title
Personally, I am thinking 1 additional champion required every 2 levels, altho numbers may or may not be right.
Edit: 6k+ IP champions can be problems. some logarithmic function it should be then.
and several other champions so by the time said player hits level 30 they will have a nice variety of 'beginner friendly picks' to work with.