What type of dining experience do you think you will want?
We have visited HH often and have tried a lot of the restaurants, from extremely casual to more pricey. To be truthful, we don't think that the ones closest to the resort aren't the best in any category.
If you're only in HH for one day and only have one dinner, it should be at one of the best places, no matter what type of dining you want.
My choices: for very casual, I would go with the Sea Shack or the Grumpy Grouper. For more intermediate, I would try Old Oyster Factory or Hudson's. For a higher-end dinner, Alexanders, Red Fish, or Old Fort Pub.
There are literally hundreds of restaurants on HHI. At the time of year you are going, some may be closed for the season, but I think you should have no problem getting in at most places.
Oh, and since you will have a breakfast there, too, try Signes! A lot of people like the French toast, but I think the fritatta is to die for!
Hope you have a wonderful experience!