Phil's Ice House (Burnet Rd, just south of Koenig Lane)
E-Z's on Lamar between 38th and 45th
Mr. Gatti's on MLK
Maudie's on Lake Austin or Lamar always has a lot of kids there
Hula Hut does too
Yellow Rose is a good spot too.
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.
I recognize that places in Austin are, for the most part, typically kid-friendly, especially those that I would consider for a family outing. However, some menus don't have viable kid options and it is nice to hear from others with experience.
There are always kids running around Freddie's, and they have a great grilled cheese sandwich. Depending on which area you're going to be in:
South: Freddie's--large outdoor area similar to Shady Grove. I believe you can throw horseshoes there too.
Central: Phil's--I must reiterate the plus of having Amy's next door
Sometimes the ice cream guys will play catch across Burnet with a bowl of ice cream to entertain the crowd
North/West: Waterloo @ 360/2222--in addition to the giant playscape outside, there's an enclosed room towards the back of the restaurant that has video games
We tried out Phil's and really enjoyed it. I thought the burgers were good and the adjacent Amy's is obviously a plus. We will definitely be adding it to our list of kid places.