Can you do Durham, or does it have to be Raleigh? How old are your kids?
Probably our favorite kid (5 and 8 year old)-friendly, casual, regional-food fine-dining spot in the area is Watts Grocery in Durham. They have a nook at the front with food-themed kids picture books, crayons, and other toys. Unlike the newspaper clipping in the window at MG mentioning someone's "least kids friendly" award, this is practically begging me to bring along the little ones :-). Although I have to say that MG always welcomes us when we show up with the entire family not just adults.
Not really sure I know what you mean by regional, particularly if you knock out bbq. Does that mean local / non-chain or are you really looking for southern food like fried chicken and vegetables, etc apart from bbq. I know you said you were not interested in The Pit but fwiw my boys really like both their bbq and non-bbq regional offerings including fried okra, fried chicken and banna pudding. Having a difficult time thinking of other similar "regional" food in raleigh---may Lucky 32 way out in Cary for more upscale. Joyce and Family in Fuquay-Varina for true home cooking.
If you are looking for mainly local, non-chain offerings, I like both Raleigh Times and Busy Bee and have been to both with my kids (ages 8 and 5) a few times. Works out ok if the kids like burgers / sandwiches etc but they are both really more bar than restaurant and things can get pretty crowded during peak times (e.g., after work drinks / diner). Our favorite place to go in downtown Raleigh with the kids is Sitti--my guys love the rice and the bread (which they call pillow bread because it looks like fluffy little pillows) and will usually share a chicken kebab plate. (Similar but much more casual and scaled down is Neomonde away from downtown over by Meredith College. ) Other places we frequent in downtown include Vic's Italian in downtown by City Market which is solid pasta and pizza but won't blow your socks off. Along those lines, we probably hit Picola Italia in Cameron Village more than any other pizza place--not really regional food obviously but a good local establishment. Both Jibarra and Dos Toquitos Centro are very good upscale Mexican that are I think a bit too fancy for my kids at dinner but may fit the bill for you as well. Other local places my kids like include Porters across from NC State, Bella Monica and Bella Mia for pizza and pasta out of downtown. They like the local Chubbys for taqueria style fare and also really like Mez in Durham for more upscale Mexican. We also sometimes hit Tobacco Road Sports Cafe on Glenwood for solid sports bar fare and atmosphere. One of their favorite places, alas, is Firebirds in North Hills which is a chain so probably doesn't suit your request here but I think they do a pretty good job all around but have a particularly good kids menu offering. Save some room for popsicles at Loco Pops if you end up eating close by.
Bella Monica Restaurant
3121 Edwards Mill Rd, Raleigh, NC 27612
Raleigh Times Bar
14 E Hargett St, Raleigh, NC 27601
The Busy Bee Cafe
7719 Highway 25 N, Ware Shoals, SC 29692
5410 Page Road, Durham, NC 27703
2025 Renaissance Park Pl, Cary, NC 27513