In this economy free has become both a loathed word and a brilliant word. Loathed because sometimes free isn't worth all the trouble you put into getting something for nothing. Brilliant because sometimes when something is free? You can actually find new ways to enjoy it rather than feeling you have to get your moneys worth.
Of all the free things to do in Indianapolis the Indianapolis Museum of Art, or IMA, is by far my favorite. Especially when I have my daughter in tow; free is the only way I'm going to be able to go. (Four year olds and art museums have a love/hate relationship with each other depending on the day.) 
The IMA is a beautiful building with hundreds of fascinating pieces of art. The clothing exhibits are by far my favorite while my daughter is most entertained by the modern art. (I believe one of her friends put it perfectly when he declared "WHAT A MESS!" in the general direction of a Warhol on our last visit.) 
No, your child's attention span may not last long in the IMA, but maybe some of it will stick, and starting them young will begin to grow a love and appreciation of art in all its various mediums. And if they hate it? It was free. So you're not out anything and plus it's air conditioned. (I've noticed since moving out here air conditioning can make or break a deal for me.)
Once you're done touring the IMA you can head outside and let your little ones run wild in the beautiful gardens of the Lilly House. A true piece of Indianapolis history, the Lilly House is surrounded by well manicured gardens thick and heavy with flowering bushes and greenery.

The Indianapolis Museum of Art is open Tuesday thru Sunday with varying hours, the gardens are open for milling about from dawn to dusk throughout the year and is located just outside downtown Indianapolis. And again, with free admission to the museum, the gardens and the Lilly House? The IMA is one of the best free Indianapolis attractions for families on a budget (and a love of air conditioning.)