The OP says that a youth gun is heavy enough to be difficult, so most of the recommendations seem to be worse than what is already available to her.
I would think that for that small a person, recoil would also be an issue.
If she's amenable to training, and will be careful as to range for a shot, the .223 would suffice. Probably the next step up would be a .243 with the Federal 85-grain Sierra HPBT. I know from experience that it works well on deer and coyotes.
My question, then, is what rifle is lighter in weight than the Howa youth rifle's 6.8 pounds?
Options: Cut an inch or two off the barrel. Use a lightweight scope, such as the Weaver V3 at some 9 ounces. Still, the gross weight would be near eight pounds.
$$$ Option: A Remington 700 Ti; the-short action version is 5.25 pounds plus ammo, scope and mounts, and sling. Mine in 7mm08 weighs 6.5 pounds, total, ready to hunt. But, nowadays, the bare rifle is somewhere around $1,100, SFAIK. Four inches cut from the 22" barrel would make for a truly lightweight carbine.