We were on the Triumph in October of last year and stayed at the Embassy Suites (Convention Center). It was great for us. We were close to Harrah's, walking distance to the French Quarter, and also able to walk to Cafe Du Monde for Beignets.
It was nice waking up and seeing the Triumph just a few blocks away. We thought about a Taxi to go to the ship but since it was such a nice day, and our luggage was on wheels, we walked and enjoyed the morning.
We rented a car one way from Hertz. After unloading our luggage at the Embassy suites we dropped it off at the Hertz location about two blocks from the hotel. When we returned from the cruise we caught the shuttle, but you can walk, to Hertz and picked up our one way rental back to Texas. No parking fees and saving the miles/wear and tear on my car made it worthwhile for us.
New Orleans is a great place to cruise from and you will love the long trip down the Mississippi.