When we are celebrating the specific occasion and when there is nothing specific to the eve or the day following the greeting, we don’t add an ’s (“Happy New Year!”).
As we know, apostrophes are used to show possession or take the place of omitted letters in contractions. In the case of New Year’s, we’re talking about possession: the phrase “New Year’s” is a common shortening of both New Year’s Eve and New Year’s Day, two holidays that celebrate the beginning of a new year. In the names of these days, the apostrophe indicates possession in that they are referring to “The Eve of the New Year” and “The Day of the New Year.”
