I LOVE NOLA and go as often as I can talk DH into going. I can definite recommend eating at Mother's Restaurant. It is an absolutely fabulous little place and the food can't be beat. It's a block or two behind the convention center and easy to find and it's not in the quarter, but in the business district.
We go downtown often when we are there, but stick to the main areas, Jackson Square and thereabouts. Don't go wandering off into the Quarter after 6 or 7 pm when all the tourists are still out.
New Years should be a pretty exciting time to be there.