I have been through your delema of choosing a rifle & caliber for an offspring in past years! My advice would be to pick the caliber in 7mm-08 from the two that you mentioned in your post.
The 7mm-08 has much larger (heavier bullets) bullets to choose from and if he were to go on an elk or moose hunt, this would be a plus in my book anyday. Nothing wrong with the 6.5 caliber or .260 Remington really, it is just more for lighter game in my eyes. I would rather hunt moose or elk with a 175 grain bullet out of the 7mm-08, than a 140 grainer from a .260 Remington.
The Sectional Density of the 175 grain bullet is .310 verses the 140 grains .287, so there is a big difference in penetration capabilities with using the heavier bullet out of the 7mm-08. This is what I choose for my daughter when she entered highschool some years ago.
List of bullet weights for 7mm-08 are as follows: 120 grain, 140 grain, 150 grain, 160 grain and 175 grain bullet weight.