You couldn't go wrong with any one of them.
I have a Model 98 in 243, it's a great rifle. I have a rifle in 308, and it is a superb round. Although I don't own a 7mm/08 I have a 7 x 57, and they are somewhat similar in ballistics and recoilwith the nod going to the 7mm/08, but just a little.

The recoil on all 3 is very do-able, with the lightest being the243, then 7mm/08, and finally the 308.

But that said, the 243 is a puzzycat, the 7mm/08 is a sweetheart, and the 308 is comfortable, as long as the gun fits him well and he holds with good form.

I'd probably say go with the 308 as it will by far have the easiest ammo acquisition and the cheapest too. And it will provide him more than ample deer killing potential as well as many, many other mid to large size game choices. Lighter bullets and a new zero could also make it a decent varmint gun, though not a perfect choice for little stuff.