Shooting at a target
When it's too dirty to spend much time outside, and your child is itching to get active, shooting at a target is just the activity he needs, and it also helps develop his eye and coordination. Collect snow and make snowballs from it, then freeze them in the refrigerator (if the snow has melted, it is better to make snow cubes). Prepare enough snowballs or cubes. Once they're hard (but not too hard—if they're too hard, move them from the freezer to a refrigerator shelf to keep the snowballs from turning into actual weapons), place the snowballs in a plastic bowl.
Have your child stand on a screened porch or other sheltered area while you watch to see if he hits a target, which could be a tree or a circle in the snow. Start with an easy goal, and if your child succeeds at it, make it less accessible.