The Rehoboth Foodie (who has an inexplicable need to refer to himself in the third person) lists the local eateries holding New Years Eve/Day parties, dinners, etc. This list is a work in progress and increases daily through the week.
Note that many restaurants like Touch of Italy, 1776 and Blue Moon sell out very early. Click here and scroll down the specials list to see the New Years activity in the far left column. Check back often for new postings.