Well if you want to go cheap I know a rifle that shoots bbs and pellets. Remington Airmaster .177 which is essentially a smooth bore barreled version of the Crosman 2100 painted black.
You can find these are Walmart for cheap prices, comes with a cheap scope that isn't all that great. It might be a little heavy and a long for his arms. If want something rifled instead of smoothbore than go with the Crosman 2100.
The lead pellets will make you handicapped.
Just btw
They make pellets out of other materials besides lead now days, it's just that lead is more common. I rarely lead when it comes to shooting since I don't want that stuff bleeding into the ground over time.