For simplicity sake, I'd go with the 243.
If you buy a 6mm Rem, make sure the twist matches your hunting application; that is the first concern. If that Ruger is 1/12, I wouldn't want it for deer. 1/10 or faster is best for heavy bullets.
Next, as others have said, ammo availability is important, and the 243 is ubiquitous while the 6 is rarer.
Another detail, again like others have said, is the action length. The 243 is SA, based on the 308/ 7.62x51, while the 6mm is a descendant of the 257 Roberts, which is based off the 7x57 long action cartridge. So while it's not a deal breaker, shorter actions are always nice when it's an infinitesimal velocity trade off.
.....So, all things considered, the 243, 260, or even 7-08 are great for kids. When my daughter was 12, I got her a 7mm-08.