re: .223 .243 7mm.08 22-250

I've killed a bunch of deer through the years and cleaned even more. Based on my experience, of the three options you mentioned, I'd recommend the 7-08. I know, a lot of deer have been killed by 22 ceterfires and 243 through the years, but I don't think either are adequate for the majority of deer hunting, especially a beginning deer hunter. Admittedly, recoil can be a problem w/ beginning deer hunters, but the modern factory produced "reduced/low/managed recoil" loads can help a great deal. A reduced recoil load in a 7-08 will have about the same recoil as a fully charged 243. When faced w/ the same decision a few years ago, I got my 10 y/o son a 308 Win. I'd suggest that as an option also as recduced recoil loads are available for the 308. Get a youth model gun so it'll fit him better. Fit it w/ a high quality recoil pad. If adults decided to shoot it, put an slip on recoil pad over the butt so the length of pull will better fit the adult. Don't try to make the gun super light as a lighter gun recoils more than a heavier gun. Use BOTH ear plugs and ear muffs when he shoots the deer gun. When shooting from the bench, use sand bags, possibly placing one on the barrel fron stock (if doing so doesn't affect point of impact). Consider a thin sand bag between the shoulder and recoil pad.

If you feel more comfortable w/ the smaller calibers, then I'd recommend 243>>22-250>223. Use a controlled expansion bullet like the 100 gr Nosler Partition in the 243. Use heavily constructed big game bullets in the the 22-250 and 223.