For ancient Roman agriculturists, the months of December and January were downtime. Despite this, January was a relatively quiet month, events-wise -- not a month brimming with festivals. Major events included the Agonia and Carmentalia. The Aedes Veiovis, temple of Vediovis, on the island in the Tiber (connected in some way with Aesclepius), was vowed on January 1, in 200 B.C., according to The Roman Festivals of the Period of the Republic, which also says it was a custom to give New Year presents, called strenae (étrennes), from at least 153 B.C. For more on the gifts, see Roger Pearse.
For more on January events, see: This Day in Ancient History