Some time in the 14th century, I think it was, a decision was made somewhere to move New Year's Day to January 1st. It had been Septuagesima Sunday up to then. You can even check this out easily by considering the names of some of the months. September, October, and November, and December ought to be the seventh, eighth, ninth, and tenth months, if you go by the significance of the words, but they aren't. They used to be, and February was the twelfth, approximately, depending on Septuagesima Sunday. Historians need to be careful about the dating of events prior to the 14th century because of this.
I haven't researched the reasons for this, but one can easily surmise that the old new year's day was so designated because it was the time of the shifting one's liturgical gaze from the Nativity to the Passion and Resurrection of our Lord. As for the new new year's day, it is sensible to count the years -- anno domini, years of our Lord -- from the Lord's manifestation in the flesh. In the West this is observed December 25th, roughly the winter solstice, though I don't know why it's a few days off.
Perhaps it's because since it's better to start the year with the start of a month, the day of the Lord's circumcision and naming wound up being observed on the day of the year's beginning, January 1st, since it is the eighth day from His birth.
Eight is a big deal: it's the first day of a new week. It's a harbinger of Easter. This day is also a harbinger of Good Friday, of course, since it's the day of observing His circumcision in the flesh, the first shedding of His sacred blood.
So therefore, here's my guess: they counted back to December 25th to observe the nativity, after choosing the first of January to observe the circumcision and name, and to observe the new year. This would also serve to suggest an origin for the practice of making new year's resolutions; rather as if to say, "be renewed by the circumcision of your mind."
Anyhow, that's my unresearched but educated guess.