At that age, they'd likely have a blast out at
Sliding Rock.
It might be a stretch, at that age, but there's always the
Biltomore Estate.
Hiking and waterfall sight seeing adventures out at
Chimney Rock, just carefully choose which trails you attempt.
If they're adventurous types, in downtown Asheville they have a rock climbing gym just for kids
Climb Max.
A short drive away is Lake Lure, which is a pretty snazzy resort lake with beaches and other activities.
Pretty much everywhere you turn there is a
Gem Mine, even a few simple $5 buckets will likely give them a bit of fun finding an assortment of gems. It's ridiculous how common they are.
I've never been there, but there's a small quirky western themed amusement park not far away called
Ghost Town in the Sky.
There's also great rivers all around for calm rafting or float-tubing during the afternoon heat. Just please use one of the outfitters, as only certain sections of the rivers are safe for it. There's just too many white water rapids and waterfalls, to go without knowing in advance where they are.