Hi Silvertip,
Are you just asking for recommendations or actually looking for rifles for sale? There is an Equipment Exchange forum on this site with literally hundreds of guns for sale. You will have to contact the mods and get permission to access the forums.
I agree with Boomer about the 7600 being a good rifle. And the Savage 99, in .300 Savage if possible. It delivers near-.308 Win performance with less recoil, and ammo's easy to find. If you'll strictly be deer hunting, no moose in future, a Marlin 336 in .30-30 is hard to beat. Relatively inexpensive, too. And lighter than the autos, except for maybe the Ruger .44 Mag.
I disagree about the autoloader. I believe a new shooter can be taught from the ground up to use an autoloader. That's the first thing military recruits use, after all, and many of them have never fired a shot.
You didn't say if your cousin has done any plinking or paper-punching. or how big he is. If you will only hunting deer and smaller, a 7600 or 7400 or BAR or BLR in .243 would be a good rifle as well.
If your cousin considers bolt rifles, his options will open up even more, but I think that unless he chjooses something very impractical, like an obsolete Remington pump in .32 Remington or something, he should get what he really wants; he'll like the rifle that much better.