A bunch of these things are mentioned below, but I will try to build and add to what's already been posted:
- WWII Museum. They did a great job with it. There's also a great restaurant inside called "American Sector" that is one of John Besh's restaurants. He is one of the best New Orleans chefs around. Both the museum and the restaurant would be fine for your kids, I think (I'm assuming the concept/history of war is not completely foreign to your 7-year old, given your job). You should get in free with your military ID and probably your spouse and kids too. If not, neither the museum nor American Sector are expensive.
- Ruby Slipper in the CBD for breakfast. Awesome take on diner breakfast, fantastic Bloody Marys, and if you order a double mimosa, you get approximately half a bottle of champagne in your glass with the OJ.
- Oak Alley Plantation - really neat look at old southern life and the place is just gorgeous. It's about 40 minutes from downtown NOLA, as I recall. You can drive there or there are tourist/shuttle buses that will take you from the Quarter. There are other plantation tours available as well, but that's the best one IMO.
- Make sure you all get to try some gumbo and jambalaya. For a pretty solid, standard gumbo, the Gumbo Shop recommendation (just off Jackson Square) is a good one. There's not too many places in town where you can go wrong with jambalaya. Also try to find some boudin balls or other form of boudin - it's blood sausage, but you wouldn't know it and it's delicious.
- I wouldn't recommend Commander's Palace or Gallatoire's with kids, but you might be OK. Galatoire's will require that men wear jackets, even for lunch - it's that kind of place.
- The aquarium is OK. My daughter (1.5) loves it and they have some cool species (like the albino gator), but it is no Shedd Aquarium or anything close to it. It can also get very crowded. I haven't been to the insectarium. The zoo is actually a really nice one. The weather this weekend may limit the animals' activity a little bit, but I went in similar weather with my daughter and had a good time about two weeks ago.
- I also second the recommendation to avoid Bourbon Street - it was fun when we were in the Sugar Bowl and I was in law school and single, but most people who live here stay away from there most of the time. I also wouldn't go out of my way to go to Mardi Gras World, but I haven't been there either. I just have a somewhat low tolerance for the Mardi Gras scene and MGW is not in a great area.
- If you can keep the kids up late enough, you can go to Davenport's in the Ritz on Canal and hear Jeremy Davenport and his band every Thursday-Saturday night. They're excellent and always a good time and crowd - he plays trumpet and sings and his music is in the vein of a mix of Sinatra, et al., and some of the jazz greats from New Orleans and elsewhere. It's a bar, but it's in a hotel lobby and I think they'll let kids in.
- If the weather holds up, you could go check out Loyola New Orleans and Tulane's campuses (they're right next to each other), and then go for a walk in Audubon Park. It's a beautiful park with a little executive golf course, some beautiful homes surrounding, various lakes and parks and such, and it's right near the zoo.
- Try your best to get to K Paul's Louisiana Kitchen. It's in the Quarter and is Paul Prudhomme's original NOLA restuarant - fantastic food and great military discount, great for kids and adults alike.
- Some other good restaurants to look into, if you are so inclined: Domenica; Luke; Boucherie; Herbsaint. Those last two might not be great for kids but the first two definitely are fine.
- Drinks: sample some Abita beers (Jockamo IPA, Amber Ale), but also try some of NOLA Brewery's options (Blonde, Tin Roof, etc). You can find a decent hurricane most anywhere, though the iconic one is at Pat O'Brien's in the Quarter. You can get into there from the side street and avoid Bourbon if you are so inclined - that entrance is right next to Preservation Hall. Try a Sazerac, and your wife might like a French 45/77/whatever they call it.
If I think of anything else, I will edit or reply again. Enjoy your visit!