The New Moon on December 31rd will be the second new moon this month.
There doesn't seem to be any special name for this second appearance.
When there are two full moons in a month, the second is sometimes referred to as a Blue Moon, though it is not blue. The name is based the historic event of the eruption of Krakatoa. The ash-clouds from the eruption tainted the moons colour with blue for months afterwards.
