First, let me preface this review by stating that I haven't been to the Museum of Florida History since I was 12 (as a part of a school's day trip). I don't really know if anything has changed since then, but I have both fond and boring memories of the Museum of Florida History. The museum's goal is to present an overview of Florida's cultural past and role in shaping the US. My favorite part of the museum is the reconstructed 11 foot Mastadon that is at the entry of the exhibit hall. I spent many days in grade school imagining living side by side with these huge animals (as the native tribes did). To be honest the rest of the museum pales in comparison. They have a steamboat replica, an old school motorhome (in an exhibit about Florida tourism, and a Ford Model T. The museum has a bunch of smaller exhibits that deal with some of the cultural intricacies....but again, I don't remember much after seeing the Mastadon. They have guest exhibits every season (Napoleon is the current theme) and they usually hang a banner on the building to advertise the current exhibit.
The bad news is that this museum lacks the interaction that other museums offer (like the caboose or aquarium at the Jr. Museum or the Science floor at the Brogan), and often fails to be exciting to children. The good news is that this is a good place to visit on a budget (admission is free with donations accepted). This also means that any kids in the leon county public school system will be bussed to this museum several times during their formative years (my count is about 4 class trips). With this in mind, I'd advise parents to either let the teacher's handle this expereience or to develop a fun activity to encapsulate your family's museum visit and make it more interesting for the little ones.