Most of the 20 gauges I've shot kicked just as hard or worse than many 12 gauges. Could be that this 8 yr old is a small framed kid and maybe a 20 would be too much kick for him.
My first gun was a 410 and it has killed many deer with slugs. Of course I picked my shots with it and never have killed a deer over 30 yds with it. I also hunted for yrs with a 12 gauge pump and can't say that lost but maybe one deer that I chose to shoot at . . . .and that was due to poor tracking on my part as I found him the next morning but lost him to coyotes.
That said, if money is'nt an issue right now there are many choices for calibers that will take down deer but not kick too much for an 8 year old. I bought my son a 243 that shot real nice. He soon put it down for an 06, but the 243 took several deer nicely.
He can do it with the 410 if you can set him up where he will get a close shot and if he can properly place his shot.