I don't understand. What do you mean by "the holiday itself"?
The holiday is the New Year's Day holiday. In some countries the 2nd day of the year is also a holiday.
If you mean the holiday period (also known as vacation) around New Year, we typically bundle that together with Christmas.
Many people stop work on 24th, or even 23rd, December, and don't go back to work until 3rd January (or 7th in Spain).
Then, what is your question? Is it about what we call that holiday period?
We tend to call it the Christmas/New Year period. You could say "My favourite holiday is Christmas and New Year".