You don’t specify what sort of genre you’re going for, so I’m assuming you want to stay mainly fantasy-based. GURPS would be one possibility you might look into. Quite a few of the other game systems (for example, Call of Cthuhlu) migrated to the D20 system D&D 3.0/3.5 was based on. The Pathfinder system (currently available as a free beta test download) is an attempt to be a viable alternative for those not wanting to move to 4th edition D&D but wanting to fix some of 3.5’s more glaring faults. I’ve also heard that Munchkin is a fairly simple (if somewhat silly) game system.

Of the games I’m familiar with, I would recommend Chaosium’s Basic Roleplaying, given your stated audience. It uses a fairly simple skill percentage system - skills (with mostly intuitive names) are given a score and you roll percentile dice under the score to see if you succeed. Mostly, your kids can be told (or decide) that they have a Climb of 42% or a 1-handed sword of 77% and you can handle the ramifications and details. Combat is fairly straightforward. The game also has a number of options (two magic systems, psychic powers, mutations, superpowers).

In my opinion, GURPS and the various White Wolf games are both more complex and more language-dependent than you’re looking for. D20 is a recent incarnation of Dungeons & Dragons, so it’s definitely not what you’re looking for. I know that there are some other simple systems out there, but I’m not familiar enough with them to give you a recommendation.