Father of 3 boys checking in.

You basically need 2 things at this age, food and fun.

Cheap food is pizza at Costco.

Free:
Mountain biking at Pocohantas or Belle Isle
Skateboarding at local skatepark
Organize a scavenger hunt in your neighborhood or a park.
Buy a couple of red ryder BB guns or youth archery kits
Play the start with a penny and trade up game

Paid but on the cheaper side:
Richmond Draftcade is pretty sweet ($10 per person)
The spring race at RIR can be done pretty cheap for kids.
Take the train to DC for the day. Tickets for kids are 50% off right now