Have you been to a podiatrist to see if orthotics are advisable? My sisters and I had them from the time we were about 3 years-old. I have to wear them as an adult to help with my flat feet/arthritis. Therefore, the shoe isn't as important as the orthotic (though our podiatrist is specific that we should use a non-stability shoe to allow the orthotic to work correctly).
You can make a quick call to your insurance to see how they cover the cost of the visit to a podiatrist (because they're recognized by the American Medical Association, it should be a covered expense) and the orthotics.
As someone with bad feet, they've been my saving grace for a few years. I had my first foot surgery at 19.
The best shoes for running in general are usually Saucony, New Balance, Nike, etc. All are available in kids sizes (though you may need to go to a higher-end department store or specialty kids shoe store). We bought our son a pair of New Balance (he's 3.5 and wears a 10.5 shoe) for $40 at Von Maur.