SalarShushan: January is not the first month or the New Year in the Jewish calendar. January usually corresponds to the Jewish month Shevat (11th month) instead of the first month which is Nissan (Passover.) The Jewish New Year begins with Rosh Hashanah on the first day of Tishrei (the seventh month, usually near September) and is said to commemorate the creation of Adam and Eve. I've never heard of any other calendar where the first month and the New Year are not at the same time.
That actually parallels the Roman calendar, which is no surprise given the overlap of those two cultures. The names of the months are dead giveaways that the Romans numbered them differently, especially since September, October, November, and December begin with the Roman numerical prefixes for 7, 8, 9, and 10. Some of the other month names come from gods (January, June) or Caesars (July, August). Even more tellingly, the old name for August was Sextilis, which includes the prefix for 6.
This system is what is known as the Julian Calendar, after Julius Caesar who oversaw its construction. It began on March and ended on February, which is why we add leap days to February (another Roman invention, as they added the extra days to the end of their year).
The Julian Calendar has been superceded by the Gregorian Calendar in most countries now, but the months retain their previous names, which are now out of alignment with their sequence.
DeMignon: As far as I'm able to recall it, the original meaning is connected to the naming of different moons. Ice Moon was the name for January (here), but before counting in months it was the name of the moon.
No surprise here too, since the term "month" comes from the word "moon". Indeed, it can be read as "moonth", in the same way number suffixes are used - "this is the
fourth day", "this is the
moonth of ice". A month also roughly approximates one lunar cycle (approx. 29.5 days). The rest are padded out to accommodate the solar cycle that forms the year. Many cultures used lunar calendars in preference to solar ones, and some religious observances are still tied to lunar events.
EDIT: Corrected lunar period as per post below.