January is named after the Roman God Janus. Unlike many of their Gods, from Jupiter (Zeus), Juno (Hera), as well as Artemis and Dionysus, Janus was not adopted from Greece. Instead, Janus appears to be of Etruscan beginnings, 

Janus is frequently depicted with two faces, one looking forward and one looking back. Janus is the god of transitions, doorways, choices and duality – opposites in fact like war and peace.

The name Janus is from a photo-IndoEuropean language that means doorway or gate.

Although Janus was not as powerful a God as Zeus or Artemis, e.g. because he oversaw transitions, he was ritually recognized at ceremonies for other Gods.

One interesting trivia tidbit is that Janus – January became the first month in the Gregorian calendar, instead of March. 

(That does explain the Biblical story of Jesus’ birth when the lambs were being delivered. That happens in the Spring, not in December and January.)