If you and the kids are in to the Junior Ranger Program the Nation Park Service has, you can get a badge at Fort Caroline. They have a nice little nature center and museum/interpretive center (with restrooms- always important to know with kids), and in October they have a Civil War Encampment to remember the Battle of St John's Bluff on October 3rd 1862. If you are in the area, check it out. This year (2012) it is the 20th and 21st of October and you can't beat free. Parking is a little scarce.
It's not much compared to the encampments and re-enacting of large battlefields, but it has a cozy feel and the kids don't have to walk far.
Outside the fort they had a small gun set up. The cannoneers were very talkative and let the kids get involved. The first was actually operated "navy-style" by a sailor.
The others were a little less ship-shape in their hodgepodge Confederate uniforms, but they were very friendly.
Rifle training commenced behind the interpretive center.
There's also periods crafts and sewing, food, and children's games. Don't forget the bug spray.