The problem with rifles light enough for thin 8 year olds to be able to carry and shoulder comfortably is that they will recoil more than most youngsters can endure, even in mild calibers like the 7mm-08. I borrowed several rifles for my tall but rail thin 15 year old son to try, a lightweight 7mm-08 with 20" barrel, a Savage Weather Warrior in .270 Win with synthetic stock and muzzle brake, and my Super Shadow .270WSM loaded with 100 gr Hornady bullets and Hogdon's youth load. His favorite was the Super Shadow with the light handloads. His least favorite was a tie between the 7mm-08 which he said kicked too hard and the Savage which he said was too loud and hurt his ears (even with plugs and muffs). My solution: a Howa 1500 Lightning in 6.5x55 Swede. Very mild recoil in a medium weight sporter rifle. Probably added another half pound with the Boyd's laminated wood stock, and while it is a little heavy to pack around, it is very pleasant to shoot off the bench and holds nicely for off hand shooting.
If forced to choose between the .260 Rem and 7mm-08 for a youngster, I would opt for the 260 Remington, but only if I was a handloader. If I wasn't a handloader, I would get them a .243 Winchester to practice with until they were comfortable enough to move up to a 6.5 or .270.