In the Church calendar, we celebrate the first day of the New Year by remembering the naming and circumcision of Christ.
Jesus was of course Jewish, and Luke 2:21 suggests his parents were sufficiently devout to follow the Law of Moses. So it was natural that they would follow his birth by naming and circumcising the new baby on his eighth day of life. That’s why celebrate on 1 January.
The circumcision of Jesus has traditionally been seen as the first time that the blood of Christ was shed, and is thus the beginning of the process his redeeming human-kind. That he could bleed demonstrates that Jesus was entirely human, like us; that (via his parents) he consented to be circumcised showed his obedience to the Law of the Bible.
Before about the seventh century, the Church celebrated 1 January as the last day of the octave day of Christmas—the ‘eighth day’ on which the child Jesus was circumcised and then received his name of Jesus.
It was probably in Spain and Gaul that it first became a ‘feast’, and therefore a celebration. These celebrations were directed towards Mary and looked back toward Christmas, so remembered Jesus’ birth.
Following the preaching of St Bernardine of Siena (1380–1444), the Roman Church changed the emphasis to celebrate the naming of Jesus. Later, in 1721, the Papacy instituted a completely separate Feast of the Holy Name of Jesus.