I worked there for an extended time recently and had a colleague who was quite the restaurant-goer. Here are the recommendations she gave me and a few observations from the ones I was able to visit. I intend to get to all of these because I trust her judgment. Note that some of these are not in the Quarter, so a taxi will be in order (you wouldn't dare drive - right?):
Geautreau's (Best scallops ever) ***(her favorite)
http://www.gautreausrestaurant.com/ Jacquimo's ***Favorite
http://jacques-imos.com/bustling, relaxed neighborhood place, so prepare for a wait. My first experience with maque choux (good)
Marti's (recently opened, same owner as Geautreau's)
http://martisnola.com/ Lillette
http://www.liletterestaurant.com/On the Magazine Street strip, which is a great area. We really liked this place.
Clancy's
http://www.clancysneworleans.com/ Restaurant Revolution
http://www.revolutionnola.com/ The Italian Barrel
http://www.italianbarrel.com/Tiny, tiny - I couldn't get in when I went.
Sylvain
http://www.sylvainnola.com/ Maximo's Grille
http://maximosgrill.com/ Luke
http://www.lukeneworleans.com/A John Besh place (can't throw a rock in NO and not hit a Besh or Brennan-owned place). It wasn't my favorite.
Patois
http://patoisnola.com/More later, I'm sure. Have fun - it's a great city.