It varies, I think, by region or franchise, but the Denny's on Fry near I-10 has either Tuesday or Wednesday "free kid's meal with purchase of adult meal" deals. I am not sure. I'll check and report back.
Could be inaccurate. Need to call before you go. That "40 Places" link that appears as a second result is listing a place named "Titos Mexican Restaurant" on Dairy Ashford, but... it's been torn down. Don't know how old that list is.
Most of the lists I've come across, especially those in Austin, are pretty outdated. They get loaded one time and no one bothers to check for accuracy after any amount of time.
Last time I checked Ninfa's had it on mondays, but they also have a clown roaming around that day, so I tend to avoid it then. Call ahead to be sure, haven't been in the market for kids eat free days for a while.