I'd find a crack barrel .22 pellet gun. That eliminates the 'see how fast you can shoot' idea. Its powerful enough to be useful even once they graduate to a real firearm. It'll give you a chance to evaluate when they're ready for a real gun.
My opinion only, in order of preference would be
1) "red ryder" lever bb gun
2) airsoft
3) Rossi single shot
B-B guns are useful training tools and act like "a real gun", put holes in paper or cans. Air soft is a bit too much like a toy, useful, but too much like a toy.Rossi makes a single shot youth rifle that accepts different barrels, start in .22 with a youth stock and switch as the kid grows.
Remember, weight is a big issue to a child, light for you may not be light for him, and just because you can cycle the gun does not mean the kid will be able to.
Don't know if you made a decision yet, but a Henry Mini Bolt is a good .22 rifle for a 5 year old. Bolt action, single shot, great sights, and a real tack driver. Bought one for my boy when he was around 4 or so. I saw a couple of them on sale at Cabela's last week.