I did a lot of research on this exact topic, including having my kids shoot several different "youth" and "compact" shotguns.
In the end, the Weatherby SA-08 compact in 20g was totally the way to go, with reduced recoil 20g shells.
There are several things you have to really consider:
1) Size (ie: being able to hold it properly with shorter arms and smaller hands) Both the LOP and the distance between the trigger and the handgrip on the stock (monte carlo or pistol grip or thumbhole) has to be smaller than usual. The comb height is also a bit different for kids
2) Weight - An 8 lb gun might have lower felt recoil but how long can they hold it up there shouldered?
3) Barrel length - 2 issues, one is that the longer the barrel, the harder it is for a child to swing properly and the other issue is that oddly enough it's easier for them to hit things with sight beads that are closer to their eyes and closer together, doesn't work that way for adults but it does for my kids.
4) Frustration factor - If they can't hit anything or if the gun is a PITA to load, bring into battery or shoot, they are gonna get frustrated with it (I'm looking at you pump action grizzly release button!) Also if it's a semi, can they work the action or is the bolt lever to small to get a good grip on for kid strength fingers? (Rem 1100 this means you!) is the safety button awkward or stiff as all heck?
5) Recoil. They are small, a 12g break open is gonna hurt and make them not want to shoot anymore.
In the end, the only way to really balance recoil with lower weight was to go with a 20g 28g or 410g gas operated semi auto
28g is a PITA for ammo and 410g is harder to hit things with (ie: pattern) and also a bit of a PITA for ammo
It was easy to get low recoil 20g slugs but... finding a semi that would work with them? The Weatherby has 2 different gas valves, one for hot/magnum loads and one for regular/light loads and cycles just fine with the low recoil (I haven't tried the extra light training stuff yet tho).
Add in the fact that you can eventually swap up the furniture for adult size (weatherby sells a kit) and swap up the barrel for a longer barrel and it really turns out to be the best choice, for my kids anyway.
Be careful and TRY THE GUN somehow before buying one (maybe you know someone who has one or something?) I tried out a mossberg "youth" 20g and the thing kicked harder than my 12g 870 (MUCH harder)
Another option to reduce felt recoil is to get a ported barrel or port the barrel