Paint your own studios seem to pop up in every town. If you don't have a local studio, Michaels art chain store offers party rooms and you could buy pottery kits as the project.
The price can add up fast with a per person fee if you have a large group. For a cheaper alternative, find a local art festival as another way to provide your kid a pottery experience.
1. Fine Arts Festival - Ridgeland, Mississippi; April - Renaissance at Colony Park includes a Childrens Creative Crafts Corner; lessons about art history and culture while kids make clay pottery, mosaics, or fiber art
http://www.ridgelandartsfest.com/
2. Wheeling Arts Fest - Wheeling, WV, June - clay learn and take project, learn slab, pinch, and coil techniques. Other classes include fiber arts and acrylic painting.
http://wheelingartsfest.com/
3. Palo Alto Clay & Glass Festival - Palo Alto, CA; July - large two day festival with over 150 artists
http://www.acga.net/cgi-bin/DJcalendar.cgi?TEMPLATE=shows-detail.html&EVENTNO=00084
http://www.clayfestival.com/
5. Minnesota Pottery Festival - Hutchinson, MN - kids tent at The Clay Coyote Studio; raku, throwing, and handbuiling demos
http://www.mnpotteryfestival.com/
6. Clayfolk - Medford, Oregon - indoor event at the Medford Armory, mostly a show and sale, but also offers a Kids Clay Area where kids can create their own clay art
http://www.clayfolk.org/
7. San Antonio Potters Guild Clay Festival - San Antonio, TX; Dec. - $5 includes museum entrance, kids under 12 free, 30 potters and gallery tours