Great choice. My wife and I gave a NEF .410 to our eldest son 7 years ago and he cherishes that gun. Learned to shoot so well with it that he “Outshines” many others at the range when the clay-birds fly. Remember several things when buying a .410: Color is only cute when their young – Please stay away from PINK shotguns!? Buy a shotgun with the best length of pull (LOP). Length of pull is how well the stock fits the shooter. Try not to adjust a stock by cutting – it will decrease the future value of your firearm. Chances are if you cannot find a “small enough youth shotgun” that your child is not ready for one.
If finances are an issue (as they often are) New England Firearms, Rossi, Marlin, all make single shot shotguns in the $90-$100 range. Double guns are expensive unless you get a deal. Next in line would be something akin to the Mossberg 500 bantam which can be had for $275. I sugest shying away from the older Bolt-actions like the Mossberg 183k et al., as they can have serious isues with blowback due to poor headspace issues. If you do decide to purchase a bolt-action, take the time to make sure it is A-OK.
Other than that have fun abnd take the kids with you. Half the excitement of owning your first shotgun is the purchase. Teach them some gun-room etiquette, etc., and expose them to the World of shooting.
Any questions, drop me a line: firstname.lastname@example.org
Good day and God bless you