[KegRiver]

I don't think there is any one cartridge that is best But know of several that might be a good choice.
Any of the smaller bore lighter load offerings should be fine.

I think 11 may be a bit young also. But some kids can handle more then others.
I gave my oldest daughter a 6.5x55 swede when she turned 15. The next year she took a Moose with that little rifle. One shot kill at that.

Now she has a daughter of her own that is old enough to hunt. But the dad isn't into hunting so I doubt my grand daughter will ever be intersted either.
Her mother still has the 6.5, but these days only uses it for dealing with Bears in the yard. That gal can shoot a tick off a Moose's nose at 100 paces.
Pitty the Bear that dines on her dog's food.