There are quite a few museums, big recreation and sport centers in Tucson area where you can take your kid. To name a few:

1. JCC

The JCC offers a number of programs and sport classes, day care, and summer camps, as well as many other activities. More info at:
www.tucsonjcc.org

2. YMCA

All of you probably know the YMCA. There are a few location in Tucson. You can find them on the YMCA site:
www.tucsonymca.org

3. Parks and recreation department

The Tucson parks and recreation department (it includes the Tucson Zoo) also offers a huge amount of programs. You can find their info at:
www.tucsonaz.gov/parksandrec

4. Tucson Children's museum

The Tucson Children’s museum may not be that big and is, to tell you the truth, somewhat primitive, but it’s a good place to spend a hot summer day once in a while. Their info:
www.tucsonchildrensmuseum.org

5) Arizona - Sonora Desert Museum

It offers many different programs for kids of all ages. They even have "Toddlers' Tuesdays"! The only minus is the location: it's a 25-30 min, drive from the center of Broadway St.
www.desertmuseum.org

6) Pima Air and Space Museum.

This museum is one of the biggest in the world. There are plenty of indoor and outdoor exhibits. It's fun for the whole family. Children 6 and under - free.
www.pimaair.org