FWIW, I started on a 20. A break barrel NEF...but he is 13 after all. Maybe just skip right to the pump and the Mossberg would be fine.
the 20 vs 12 gauge thing...well when I started using the 12 as a kid (around 11-12), I noticed the difference in recoil, yes. But it wasn't bad. Shotgun recoil, to me at least, is more of a push than a sharp kick. It all comes down to the way the gun's set up, really. Imean, don't go having him shoot 3" slugs right off the bat, but for a 13 year old, I don't think 7.5 and 8 shot on a well configured gun is too outside the realm of manageable. I hate to say it, but if you're looking for a low-recoiling gun, the Mossberg isn't it. Others mileage may vary, but the 870's I've owned, especially the ones with metal trigger guards, have always kicked a bit less due to the steel receiver. I've heard of people filling up the stock with weights, though I don't know exactly how it's done. I never used any of that - I got started early, shooting my bb gun daily and 22's every weekend from the age of 5 on and given a 20 ga. @ 9 years old, shooting all manner of buck, birdshot and slug. So there's merit to trial by fire too. So all in all I'd probably suggest the heaviest gun your son can hold and point, with light loads and a good recoil pad.
If you could find a good used Wingmaster 12 and a reduced LOP stock set, that might be where it's at to be honest. Funny hearing me tell you to start a kid out on a Wingmaster, but hey. My first 12 was a Winchester Ranger with a good recoil pad. Loved it then, would pick up another one now, for the right price.