I traveled there for work about 2 weeks per month for 3 years and I really miss New Orleans!
Here are my favorites:
Jacques-Imo's (get the alligator cheesecake). There's a trolley stop not too far from the restaurant and it's fun to take the trolley.
Acme Oyster - definitely touristy, but worth it. Try the chargrilled oysters - they are fantastic! There will be a long line for Acme, but it moves fast and you can also enjoy a beverage while waiting in line. Felix's oysters across the street is also good if you don't feel like waiting .
Red Fish Grill
Palace Cafe (this is on Canal St and it looks touristy too, but the food is great)
Welty's Deli - this is a lunch place and not many tourists go here - mostly business people (It's in the CBD). They have a daily lunch special that is great and huge portions of food. There are plenty of other options on the menu if you don't like the daily lunch special.
Rouses Grocery store is good. Lots of good options if you get tired of eating out all week.
Barcadia is a fun spot for burgers and lots of old school arcade games, skeet ball, huge jenga games. Depending on the makeup of your group, may or may not be a hit.
WWII museum is definitely worth checking out.
Frenchmen Street - I loved D.B.A., Three Muses, Cafe Negril, The Maison. All great music spots and most didn't have a cover charge. Can't go wrong with any of those places.
Have a great trip!