If you are looking for something extra special for your child, you might try Pizza King in Avon, a westside suburb of Indianapolis. It's a very casual and inexpensive place with good pizza and a variety of other stuff (I don't know about beer), but it has a model train inside a glass case which travels along the booths and brings your drinks to you. I used to take my son there for his birthday every year at his request, although I live 45 minutes away. They have video games you can play in your booth as well. My kids absolutely loved that place, which is maybe 15 minutes from the airport.
If you want something closer in, the Spaghetti Factory in downtown Indianapolis is an inexpensive place which offers good food. Not really a "can't miss" place, but my family likes it.