Cold Moon
The plainest name in the list, and it comes with the best astronomy. A full moon is opposite the Sun, so when the December Sun is at its lowest the full moon is at its highest — the December full moon rides higher and stays above the horizon longer than any other in the year, sometimes for more than fifteen hours at northern latitudes. Long Night Moon is the more accurate of its two names: it is not only that the nights are long, but that the Moon is up for nearly all of them.
Where the name comes from
Moon Before Yule places it in the Anglo-Saxon calendar, which is a genuinely separate tradition from the North American names it now sits beside in the same list. Twelve names, at least three distinct cultural origins, presented as one set — which is the whole problem with the set in miniature.
Also called Long Night Moon, Moon Before Yule.
Old Farmer's Almanac; the high-December-moon geometry from Meeus ch. 13