Pope Pius XI instituted the feast of Mary, the Holy Mother of God, on October 11, 1932. Eventually this feast was moved to January 1st, to replace the feast of the Circumcision of Jesus, after the liturgical reform of the Second Vatican Council. This change was meant to highlight the link between Christmas and the divine motherhood of Mary.
Since the Council of Ephesus in 431, the Church venerates the Virgin Mary under the title of Mother of God (Sancta Maria Dei Genitrix in Latin, Theotokos in Greek), so this solemn feast celebrates the mystery of the Incarnation at the same time.
Though the title of Mother of God is not found in Scripture, it has an ecumenical dimension: for Luther, the title of Mother of God was an absolute reflection of the Blessed Virgin Mary’s dignity. "A woman who is mother of God" is a bold formula that truly conveys the solemn meaning of Mary’s dignity, the meaning of the feast of Mary, the Holy Mother of God.
So we begin the new year with the Solemnity of Mary, the Mother of God.