It's just a restaurant, and a not-particularly-great-one at that (although not bad for a chain), but Texas Land & Cattle has "kids eat free" on Saturdays until 4 PM.
Phil's has great burgers. Waterloo has the biggest and safest kiddie play area.
Freddie's Place (South 1st) is very kid-friendly.
As far as I'm concerned, most "hip" Austin restaurants are. Guero's, Artz Rib House, the Salt Lick, Chuy's, Maudie's, Nuevo Leon, Romeo's, none of them seem very stuffy. Enjoy yourselves.