I bought a Crossman with the 6 inch bbl. It looks like a Python. I use it in the back yard to practice basics. I've also "taken" two citrus rats with it. As I recall, it was less than $50 at Wally-World.
Don't forget the safety goggles.
How old is the son? This may seem kind of odd, but this dictates somewhat what can be recommended.
Airsoft guns would make for a somewhat safer all around rifle or pistol. Plastic BBs are reuseable.
I have a Crosman 760 pump rifle. Shoots regular BBs and lead pellets and is a single shot Pumps are 1-10 depending on how much velocity one wants. This is small and short enough that a kid can hold and handle it.
Crosman also makes several single shot pump up or CO2 pistols that that would fill the bill. Model numbers change often enough that I can't specify. These are .17 caliber or .22.