We liked Queen Anne's Revenge in Wanchese (just south of Manteo). Great seafood, very nice staff, sort of a rustic looking place. If you're into breakfasts, we ate three or four times at Stack Em High. There are at least two locations on the Outer Banks and maybe more. Great pancakes (especially the chocolate chip ones) and waffles with fresh strawberries. Large selection of omelets. Our absolute favorite, though, was Petrozza Deli & Pasta Shop in Kill Devil Hills. Don't be put off by the fact that it's in a strip mall by a Food Lion grocery. This is great Italian food, homemade pasta. Unfortunately we didn't find the place until half way through our week, but managed to make up for it! We stayed close to Duck so that's why these recommendations are a little ways from Avon, but not too far.
North Carolina