Our 10 yr old now has a surprisingly sophisticated palate but like many children doesn't really care for really spicy items. You may want to try asking for a smaller portion. He loves the 1/2 portion of lamb shanks over mashed potatoes at Stoney Knob in Weaverville. Sharing an entree also works for us. Some of his favorites are: Stoney Knob, Tomato(West Asheville), and Corner Kitchen.
I have a 10 year old that love salad, shrimp, calamari, ribs, etc. I've found that splitting my meal with her is easier than trying to find something inexpensive as most restaurants worth dining in for a food in Asheville don't really have a kid's menu and when they do it is still grilled cheese & chicken fingers. Or the other option is ordering off the appetizer menu for her and splitting my salad with her.