My initial thought for your son (13 years old) in a LONG action, but with either a mag or a non-mag bolt face (since you have access to either) was the .
25-06 Rem. Mild recoil, flat shooting, and a great hunting round. But I don't know what "long range" means to you and your son (500 yards? 800 yards? 1200 yards.)
When you said you were down to the .257 Weatherby or the .264 Win Mag, then I favored the .257 Weatherby since you don't currently have any .25 caliber rifles. It would 'fill in' the gap for you nicely. Plus it's great for hunting out to 600 yards (deer/antelope; need to be closer to shoot elk and moose.)
But if LONG RANGE is a higher priority, or BIGGER game is on the menu (elk/moose) then the .264 Win Mag will provide higher BC bullets and seem the 'better' choice for a long range hunting rig.
I did like the suggestion
@Andrew Massi made earlier to consider the
.240 Weatherby if you were really interested in a 'Bee cartridge. The recoil would be lighter than anything discussed so far, but it would be a laser on deer/antelope sized game and, if twisted correctly, it can fire some very high BC bullets, too.
At 13, I sure didn't want to get beat up every time I tripped the sear on my rifle. I think the .240 Weatherby would be my 'pick' (assuming I can't 'sell' you on the .25-06 Rem still!)