We ate twice out in Nassau while we were there.
First..for lunch we ate at Conch Fritters. The menu is moderately priced and the food is fantastic. It is a block or so away from the straw market..across from a very nice hotel. Easily walked to. It is on the second floor, big open air windows, great for people watching. We (hubby, myself and daughter) loved it and can definately recommend it.
Then, when we were heading back to the ship for our Palo reservation, we noticed a filipino restaurant..right to the left of the straw market, oceanside.
Well, we love filipino food, especially lumpia so we stopped in and ordered two batches. It was made fresh while we waited and it was absolutely wonderful.
There is also a place called graycliff (like the great cigar), we found out about this after we returned. Next time we will look for it.
There is also a hard rock cafe and a senor frogs. Plenty of options..don't be afraid to ask the locals for advice on where to go. That is how we found Conch Fritters, from our tour guide (we had done the two hour historical city tour right when the ship docked..also fantastic by the way).
Just go and have a blast..why would you want to eat on the ship..when you can explore a new place.