The Louisiana State Museum is housed in the Cabildo next to Jackson Square, and I found it surprisingly fascinating. The buidling itself has played its part in the history of New Orleans, making the exhibits themselves especially evocative, since some of the events they depict actually took place here.
The museum begins with the original inhabitants of the area and what life was like for the first settlers, using everyday objects to bring it to life. The exhibits follows the history of New Orleans as it became a city and the changes it experienced under Spanish, French, and English rule.
Most interesting are the cultural-history displays detailing old New Orleans society, from dueling to the quadroon balls to mourning rituals. The museum also has Napoleon's death mask.