Janus was the god of crossroads, openings, that sort of thing. He was typically depicted with two or four faces, one looking forwards, one looking back, or in all the directions. If January is the first month of the year, then it would make sense for it to bear the name of the god who looked both in the past and the future - on the threshold.
There's a great poem by Jorge Luis Borges that mentions him called "Limits:
There's a great poem by Jorge Luis Borges that mentions him called "Limits: