Do you reload? If so then cartridge won't matter. Youth loads are easy to develop for just about any cartridge. Gun fitment is most important.
6.5G is a really good option. Especially in an AR. But the Howa mini is a great little rifle for a decent price. The Howa is not a true youth rifle, but does have a shorter LOP than most. I have both for my kids, and I wouldn't hesitate to use the cartridge on deer up to 200yd myself. I started my kids out on the AR for its collapsible stock, then I move them into bolt actions when they get bigger. Remember, lighter and shorter guns recoil more than longer and heavier ones, so a heavy barrel AR has very little recoil, but an M4 configuration may have quite a bit to them.
6.5C may be a little much for young kids just starting out, but a fine low recoiling deer cartridge with the proper loads if you find something in a rifle that fits them. Use 120gr deer loads for recoil.
.243 will also work, but I'm not a fan of it for personal reasons. I won't put it down, but I feel that the 6.5C is close enough in recoil with greater frontal diameter and heavier bullets that I would lean that direction if I was going with a "full action" rifle. The main thing is getting a rifle that fits them. (Notice a theme here?)
Most of all, keep their shots behind the shoulder with any of these unless you are using a stout bullet. A quartering-to shoulder shot is great if you have a bullet that can get to the vitals, but can be a mess if the bullet blows up. That really goes for any cartridge, but even more so for the smaller ones.
I love getting kids started in shooting and hunting. I also teach youth shooting sports to beginners through 4H. It's better to get them started right than to try to fix bad habits after they have already been started. Don't push them too fast, and if they're not ready then let them wait until they are: each kid is different in their growth and development. Feel free to PM me if you have questions.
LOTS of dry firing. Probably 5:1 or better over actual shooting. Good practice that keeps them from developing a flinch.