All the calibres mentioned would be good choices, excluding the 30/06. The .243 is the lightest calibre that is (as far as I know) legal for big-game everywhere in N America (yes I know some states/provinces allow .22 calibre centre-fire, but they're the exception). And as was mentioned, it is the definite light-weight of the bunch. It will work on deer buts gets real iffy for anything with tougher skin and bigger bones.
The only one I would add is the lowly 30/30, especially for a new hunter. Its low recoil, rugged, dependable and will take down any big game animal in the lower 48 (and most of Canada) out to 75-100 yds.
Also, bear in mind its almost a given that within a few years once they get experience they'll have their own rifle/calibre preference, which probably won't be the one you chose for them, however suitable it was. Your carefully chosen rifle will become a back-up weapon for their own; used rarely in the field and (hopefully) valued more for sentimental reasons that its field abilities.
30/30s are cheap