One of my favorite off-beat places to visit is the Backstreet Cultural Museum in the Treme.
Oh, that's one of my favorite places in NOLA! I love that museum. The Indian costumes exhibited here are breathtaking – it's hard to believe how many hours of work went into the making of these beautiful pieces. It's worth going just to see these alone. I didn't realize the owner had died - the last time I was there was probably 7-8 years ago, and he gave me a tour of everything. Speaking of Treme, if you have a chance to watch the show "Treme" (streams on HBO but others have it for rent), it was a wonderful series about New Orleans and the people there.
I second the rec for Frenchman St. for music - there are some wonderful musicians there and in other venues in the Marigny, like Snug Harbor, d.b.a., Blue Nile and the Spotted Cat. It's a fun district - it used to be much less touristed than the French Quarter - not sure if that's still true. And yes to trying to hear Kermit Ruffins play! He is amazing!
One of the best meals I ever had was at Bayonna, owned by Chef Susan Spicer, on Dauphine St. I ordered the veal sweetbreads with lemon caper butter. It was the first time I'd ever eaten sweetbreads and this dish blew my mind. I still think of it and the incredible meld of flavors. In fact, I remember that dish more than most others I've had over the years.. That will be the first place I eat when I go back. Also make sure you get a Po Boy - there are lots of places that have good ones. And I will also second the rec for Cafe du Monde beignets - oh the joy of eating one of those!
Definitely take a walk in the Garden District - this is one of my favorite things to do when I go there. You can get in some celebrity house watching if you like. John Goodman has a house in the district and lives there primarily; so does Sandra Bullock although she's not there much. But the houses and gardens are gorgeous and well worth a visit. It's a fun place to walk and get away from the raucousness of the French Quarter. Also don't forget to visit the cemeteries - there is so much history and atmosphere there. My favorites are St Louis Cemetery No. 1 and No. 3 and Lafayette Cemetery No. 1. There are tours that go through them also, but I like walking through them by myself.
Your post just made me realize it has been too long since I've been there. I use to go every 3-4 years but it's been a while. Have a wonderful trip - I think you will love NOLA!