Jackson Square, then get the hell out of the French Quarter. Unless you want the kids to see drunk redneck tourists pissing in the streets. Everyone should see it - just once. Please don't let it ruin your opinion of what is a fine city.
The WW II Museum is awesome.
You may consider the Art Museum. It is in City Park, which is wonderful.
Check out the Ruby Slipper in the CBD for Brunch - it's tremendous.
Try to get to Parkway Diner for the best Po-Boys in town.
Try some Blue Dot Donuts - it's run by 2 cops. They have an awesome PB&J donut, which I'm sure the kids would love.
A fun place to dine is Ye Olde College Inn - it's right across from the Seminary, not far from Xavier - they have fabulous food and it's not real expensive. Plus, they carry some great wine from a little grape juice factory I know very well.
Cash, Aces-Up and Bolcar47 are much more educated on the City than I am. These are just some observations I have after 5 trips there in the last 1.5 years.