It's going to be a little difficult riding around Va with a kid on his own bike, that takes the "dual sport" aspect out of things. I know of very few offroad-specific areas open to the public in Virginia. Most of the riding (including really rough roads like Shoe Creek) are only open to street legal vehicles. Only thing I know to check out is the Terrapin and Rattlesnake trail. However, since it's "one of the few" offroad only trails, it's kind of crowded and scary.
You can see them here: