This is an AmE expression; in BrE we'd say 'the New Year', which is equally ambiguous. It doesn't specifically mean the Eve or the Day, but presumably they would stay the night and 'see the new year in'.
Thanks, Entangledbank, that must be why I've not come across these. I would have assumed New Year's Eve, since on a matter of logic that that's when you stay up late. You're too hungover to want to travel on January 1st and anyway there are no trains.
I would say that both were for New Years eve. New Year's day is a day of recuperation and rest, not so much for visiting. But it remains ambiguous without more context.
New Year's Eve, n. a night that restaurants over-book, over-charge and provide lousy service; patrons over-drink and behave badly, and thus a very popular night to celebrate.
The second may depend on whether Aunt Samantha lives nearby or not (or some other context information).
If she lives nearby, you might be there for the New Year's Eve party and then go home when the party ends.
If she lives faraway, you might spend the night thus being there for New Year's Day as well. Perhaps you might arrive a few days earlier and remain a few days afterward and still refer to that as New Year's (your New Year's vacation/holiday).
Technically New Year's Eve is Dec. 31 and New Year's Day is Jan. 1--but if I ask someone if they have plans for "New Year's", I'm asking about New Year's Eve. Because generally no one really plans anything special for New Year's Day.
I think it's fair to say that "New Year" refers to the beginning of a year
December 31 is when people go out. The point of the revelry is to stay up until the clock rolls over to January 1.
January 1 is the Federal/bank holiday. Which is a good thing, because everyone stayed up late the night before.