Being that he is a youth, I wouldn’t recommend a lever. He may not be able to #### it consistently, if at all. Start him out with a bolt action rifle and go from there.
To get something decent you don’t have to spend a fortune. A used Remington 700 and try and find a youth stock, or a youth 700 with the spacer kit and change out to a better stock when he’s grown enough. Longer barrel like a 22 or 24” if possible.
I started my son with a Weatherby Vangaurd S2 in 243 before he was legal hunting age and he loved it. But he moved on to different tastes quite quickly and showed he could handle the recoil very easily, shooting my buddies 7mm Rem Mag at 12. He normally shoots my 308 Kimber Hunter now. He’s 14 now and about 170lbs, no problem with recoil.