There have been several suggestions for lunch at Mother’s, and a "Ralph" or "Ferdie" will fuel you up for a full day. BUT–don’t overlook Mother’s for breakfast.
Uglesich’s is my favorite anywhere, and Anthony is a great guy to talk with when he has some spare time. I have eaten at most of the places suggested in this thread, and New Orleans has never disappointed me.
There is another great small restaurant between Jackson Square and the French Market called "Irene’s" (I forget which street), and no one has mentioned "Bon Ton" or "Central Grocery".
For "adult beverages", try Cooter Brown’s (just about 100 yards from the Camelia Grill where the trolley turns off St. Charles). They have the largest selection of beer that I have ever seen. The array of draft choices is unbelievable.
Favorites in New Orleans? Just walk into any place on any street and you will get a treat. Be sure to get to the places the Roadfood books have reviewed. Most of them are institutions.