Serious Eats - seriouseats.com

I agree with the other comments, if you have to choose between May and June, go with May. I've lived here all my life and I don't think I ever really get used to the heat and humidity of summer.

But on to restaurants. Many good suggestions thus far. I'd like to add a few.
MiLa (milaneworleans.com)
Willie Mae's Scotch House (best fried chicken you will ever eat)
Dick & Jenny's (dickandjennys.com)
When it's hot, nothing beats a refreshing Pimm's Cup in the Napoleon House courtyard.

Finally, if you are here for a Sunday evening, I can't stress enough what a deal the Chef Sundays are at Bacchanal are (bacchanalwine.com). Buy a few bottles of wine from the shop, sit out back in the courtyard, and enjoy whatever the chef that week is preparing. Just check ahead to see what's on the menu and who's cooking for that week. I've had anything from authentic Peruvian cuisine, to a BBQ Pork "sundae" to the tastiest vegetarian greens I've ever eaten.

The worst definitely had to be the day the chef was having a really bad day and was incredibly pissed off (I can't remember at what). I was working the grill/griddle and when the order came in for some grilled chicken sandwiches, he threw the breasts at full force on the grill, splashing hot grease all over my face and neck. I was quite the attractive guy for the next month.

Then there were the deli slicer fingertip injuries, knife injuries and my favorite was that I developed severe excema on my hands from all my years in the kitchen and pretty much couldn't touch anything (citrus, tomatoes, dishwater, cleaning supplies, you name it) without searing pain for about 3 months. Good Times.

Tactful_Cactus beat me to it.

So after she asks Sandra Lee "Why?" i think I'd need to follow that up with a "No, seriously, WHY?"

Around when are you planning on coming? There's always some sort of festival going on and many are centered around food, so knowing a date would help. For instance, June 13 - 15 is the creole tomato festival and the seafood festival is June 7-8.

I second many of the recommendations above. I would also like to add in MiLa, a newish restaurant located in the CBD. (milaneworleans.com) It's southern and local classics made with locally sourced fare.

Also, you might want to check out Bacchanal. (bacchanalwine.com). It's a great wine shop in the Bywater with a courtyard out back. On Sunday afternoons, there's a local chef serving up delicious food for ridiculously cheap prices. And you can just buy a bottle or two of wine and get some glasses from inside. It's a great way to wrap up a weekend.

I am a bit confused by the $10 limit myself. If I remember correctly, they actually made six servings of food. There were the four plates served to their child sous chef and the three other kids at the table, plus the one served to the judges, plus the one Tom ate in the kitchen. I don't even think you can buy six portions of chicken at whole foods for ten bucks.