Internal Gear Hub for 16in Wheel?
It's pretty easy if you have a spoke cutter and are willing to run a funny spoke pattern. I suggest buying (ideally second hand) a three speed hub and lacing it into the 16" rim yourself. With a wheel that small and a child on it there's not enough load to make strength an issue, so you can reasonably either lace a 20 spoke rim to a 36 spoke hub using whatever holes are closest, or ignore two spoke holes in the rim and use every second hole in the hub (18/36). It's probably not going to be pretty, but it will work.

However I think you're going about this backwards. Why are you riding faster than he can? Much as you wouldn't suggest stilts so he can keep up when you walk, gears don't address the problem that you can ride faster than him. He doesn't have the power or endurance you do, so without a power assist he's going to be slower.

If you need to cover long distances, perhaps a trailer bike would be a better solution? That way he can "help", and you can go whatever speed you like. That's generally safer for your kids, since you're in control and the only real risk is them jumping off the bike while you're moving.