Lunch with the kids in NOLA today.

If you want seafood, try Grand Isle.

Not N.O. food in any way, but Lucy's has food on the cheaper side of things and they do a little bit of brunch food too on Sundays. I have taken my kids there a few times when I needed an easy meal while out with them.

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Around the aquarium, within walking distance, both kids under 5.

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Kids under five, by Aquarium I would say.
(well depends on price point)

Palace Cafe is right there and is not really expensive and good at accommodating kids and will offer a good NOLA menu.
New Orleans Hamburger and Seafood is good for a quick lunch.
especially with kids.

You could take the streetcar or walk the riverfront and go to the places closer to the French Market. A lot of choices there that are family friendly also kids like the area seeing boats, and streetcars pass.
I would say walk around and look at menus in that area and step into a place that has a dish you would all want.