It's New Year's Eve -- all day long

Every year I seem to have an argument with somebody as I point out to them that there is no such thing as "New Year's Eve day," or "New Year's Eve morning."

While many like to think the eve part of the phrase denotes evening, it is in fact the entire period, or day, before New Year's Day (also applies to the Christmas holiday).

From my Apple dictionary:

eve |ēv|
noun
the day or period of time immediately before an event or occasion : on the eve of her departure he gave her a little parcel.
• the evening or day before a religious festival : the service for Passover eve.

So please just leave it at New Year's Eve. If you want to be more specific, say New Year's Eve at 12:33 p.m.