January 1st in the Church is not the celebration of the New Year, that would be the First Sunday of Advent, liturgically speaking.
On January 1st we celebrate Mary and her role in salvation history – being the Theotokos, the God – bearer, to use the theological term. This title was confirmed at the third ecumenical council, the Council of Ephesus, that was convened in the early 5th Century in response to the Nestorian heresy.
“Mary’s divine motherhood was proclaimed at the Council of Ephesus in 431.
Various names are used to describe Mary’s role as mother of Jesus. She is called “Mother of God” which translates the more accurately stated greek term “Theotokos” or “Birthgiver of God.”
The Council of Ephesus (431) attributed to Mary the title, Mother of God. This needs to be read against the Council’s declaration that in Christ there are two natures, one divine and one human, but only one person. Indeed, according to the Council the holy virgin is the Mother of God since she begot according to the flesh the Word of God made flesh. This decision was further explained by the Council of Chalcedon (451) which says with regard to Mary’s divine motherhood:
‘…begotten from the Father before the ages as regards his godhead, and in the last days, the same, because of us and because of our salvation begotten from the Virgin Mary, the Theotokos, as regards his manhood; one and the same Christ, Son, Lord, only-begotten…’
Mary’s Divine Motherhood was not the object of an independent or exclusive dogmatic declaration. The statement is embedded in texts defining the person and natures of Jesus Christ. Thus, the dogma of Divine Motherhood becomes an integral part of the christological dogma. This does not diminish its definitive and binding character. The dogma of Divine Motherhood is generally accepted by all Christian denominations.”
Source the Catholic News Agency: https://www.catholicnewsagency.com/resources/mary/general-information/the-four-marian-dogmas
For those seeking more information on the Nestorianism heresy, here’s a link to an article on Catholic.com : https://www.catholic.com/tract/the-great-heresies
For those seeking more information on the Council of Ephesus, here’s a link to an article on Catholicism.org : http://catholicism.org/the-council-of-ephesus.html