This isn't as much opinion about the best youth shotgun as it is my actual experience this year. I tried to force things in a different direction but wound up very satisfied with the way things have worked out.
I just started my 8 year old son, Pierce, shooting a shotgun and dove hunting this year. Over the last winter and spring, I had him hoist a lot of youth size shotguns to see how he handled them. What became the determining factor for me was getting something that was not too heavy for his left arm to handle. He shoots right handed, is 4'6" tall and weighs 82 lbs. The Youth model multi shooters were all too heavy in the fore end. I had him hold them up as long as he could. I did not time it, but his left arm got tired way too fast for my liking. (Maybe, 5 to 15 seconds, depending on the shotgun, and he started wavering considerably). I have to admit I felt a little disappointment that the multi-shooters were not going to work out. (disappointment in this case is correctly spelled b-r-u-i-s-e-d e-g-o)
I got my old Ithaca Mod 66 Supersingle out that I bought when I was 11 years old for my first gun. The length of pull was 14" and obviously too long (no recoil pad, just a plastic butt plate) and it too was a little heavy in the fore end. I bought some 7/8 oz. 20 ga. loads and we went to the range and shot some trap. He got beat up pretty good. We went home and cut 2-1/2" off the stock, and installed a 1" recoil pad, http://www.lymanproducts.com/pachmayr/, Pachmayr Youth Decelarator Model #01380. He handled the gun a lot better and had much better fore end control. Back to the range, to shoot some trap. He loaded a round, called pull, fired, the clay bird broke then he looked up at me, smiled and said "That didn't kick any harder than my BB gun". It still kicks pretty good because I see his shoulder and head recoil with each shot, but his perceived recoil is minimized enough that he is now having a lot of fun shooting it and never mentions recoil.
We went to the trap range several times in August, and it was pretty hot. He would shoot 5 shells and we would take a little break, then 5 more and another little break. He has been drilled in gun safety (with a 45 year old Daisy BB gun) since he was 4 and does a very good job. The 5 rounds and rest thing gave me a good opportunity to watch him transition from shooting to racking the gun then back to shooting in safe manner. It also seemed to keep him interested and fresh. Some of the old guys sat and watched him shoot and their first comment was always, "Your doing a great job with gun safety son, keep it up". I was surprised as he regularly broke 3 and 4 out of 5 targets, from straight behind the trap with the wobble turned off. I did notice that when we tried to stay at the line and shoot 10 or more rounds without a little break that his left arm began to tire and he started missing more. When his left arm gets tired, I notice that he begins to arch his back and shift his weight to his rear foot, it's time to go right then and not push it. As I recall, the most he has shot at the trap range in one outing is two boxes of shells.
"Braggin part" - We went dove hunting and on opening day, he hit his first bird on the wing at about 15 yards moving about thirty miles an hour. I was trying to get the word "wait" out as the bird was too close. I did not speak in time, and he took the head clean off and did not put a hole in the eatin' part. We have gone every weekend until today, and he has shot 2 to 4 birds each time. We are both having fun, he is very happy and safe with the gun and enjoying success. He has already commented that I kill more birds because I have more chances with my pump shotgun. Now his appetite is whetted to move to a more-than-one-shooter so we have another "new gun experience" to look forward to.
Long story I guess, but it is my actual experience. My opinion based on this experience, is
1) Be sure the gun has a LOP that will let the child mount it properly and light enough for the young'un to hold it up, as if pointing at a bird, for 20 seconds or longer. This obviously has a lot to do with the size of the child and the weight distribution of the gun.
2) Put a high quality very soft recoil pad on the gun. (I paid $50 for the Ithaca in 1968, and $30 for a recoil pad in 2007) It is worth every penny when you see the difference in your childs reaction to the recoil.
3) Get light loads to reduce recoil. (Pierce has had no problem breaking clays or killing dove with AA TARGET LOAD 20 ga., 2-3/4", 2-1/2 DR.EQ., 7/8 oz., #8's or #9's. The gun has a modified choke and has worked well on dove at 35 to 40 yards.)
4) Take them too the trap range or get a clay bird launcher and let them shoot at their own pace. Have them take a break before they appear to be tiring. Give encouragement, congratulations and safety commentary only, no shooting advice unless you are an expert shooting instructor. They will figure out lead and timing on their own, it's amazing, and will enjoy it much more that way in my opinion. ( My wife was kind enough to explain this in detail when she started shooting with me )
5) I had my head set on getting a youth multi shooter in 12 or 20 ga. and shooting light rounds. However, after this experience, I really feel the single shot has been the best thing for both of us.
I don't want to preach to anyone about Safety, simply offer my personal veiwpoint. With the break open action, it is very easy for me to see from a distance if the gun has been made safe. I have stayed within arms reach of my son at the range and on these first few hunts. When I let him go pick up a downed bird, I follow close behind. As I get moreconfidence in his continuing Gun Safety, I am sure I will let him begin to get a little farther away from me. If, for any reason, I have to tell him from a distance to make his gun safe, I know I will be able to see the action broke open. My pride wants to say he would be safe with any gun, but my common sense says I need to take every pre-caution I can and the single shot is one of those pre-cautions for me.
Good luck, searching for the right gun is fun, and watching them have fun with it is even funnerer-er.