See if you can get an HR Ultra 20ga slug gun with an inexpensive scope should run around $400-450. Use 2 3/4 Remington Accutips. The weight of this gun takes the recoil out.
Just checked the sighting on ours today, my son's last youth day is Saturday. He has dragged "Bertha" as in Big Bertha all thru the woods since he was 10. Neer missed with it. Hit the 1 inch bullseye in one shot today.
May be easier just to find a youth stock and throw it on one of your guns for now to get him through this year, then you'll have more time to find something nice for him over just getting what you can find.
If by chance you have an 12ga 870, I have a black synthetic youth stock for one I don't need, and will be going to browns mills in a couple hours.
It will fit an 1100 also, and think the 1187 to.
To help a youth out, you can have it as a freebie, I just ask that you pass it on, when its no longer needed.
Seems kind of late to be rushing into a cheap gun so you can make youth day in 3 days. Just seems to me like you would want to get him something and get him out shooting so he is safe and comfortable and confident to make an ethical shot. Plenty of sport shops with used guns to cut down on price without just picking up the first honker that comes along.