I'm just joking. There may be a better cartridge for what he wants to do, but the 7mm-08 is a .308 case necked down to 7mm, so there really isn't anything between them. There may be a few ways to tame the recoil on the .308, making it easier to shoot..
If its just recoil thats holding him back from a 308 try an aftermarket stock. My 308 varmint barrel used to kick like a mule with 180's untill i put the MDT ORYX on it now it recoils like my light little 223 savage
Reduced recoil rounds work great in my Grandson's rifle, he shoots a 7mm-08. The bullet is designed to expand at the lower velocity of the reduced load. Try to find some reduced recoil .308 ammo, or have him shoot a mono bullet in the 125-130 grain range.
It would help us if we knew a little more about your son. How old is he, what experience does he have, and his reason behind wanting "bigger" than 7-08? My daughter shoots a Howa Alpine in .308 usually 125 grain Nosler NBT or AB bullets, but she can handle up to 165 grain bullets pretty well from a 7 lb all up rifle. She's been shooting a .308 or .300 Savage since she was 11.
I would stick with the 7-08 as it has proven to be a very good game round. With proper loadings, it comes very close to 270 winchester performance and being a .284 bullet, there are many options available. I load the 120 Barnes TTSX for deer sized game at 3100 FPS for myself and two of my friends and it has done a very good job on deer and pigs. And i am working up a recipe for the Barnes 145 Long range bullet now for bigger animals. One of my 7-08s weights 7 1/4 lbs with scope, sling and ammo and the other weighs 6 1/4 outfitted.