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Little known fact: The Ordinary Form of the Roman Rite celebrates three things on the January 1 octave of Christmas: Mary, Mother of God; the Circumcision of the Lord; and the Holy Name of Jesus. The first takes precedence, but the other two are also officially commemorated.
The Roman Martyrology of the Extraordinary Form simply commemorates today's feast as the Octave of Christmas. The OF Martyrology, on the other hand, restores the official commemoration of the Circumcision and explicitly lists all three aspects in its entry for today's solemnity.
The Circumcision and the Holy Name are featured in the readings specifically. The Gospel reading, of course, recounts taking the child Jesus to the Temple on the eighth day. The First Reading from Exodus is focused on the Holy Name.
The Second Reading from Galatians (which is an excellent OF addition, not found in today's EF Mass) hits all 3 elements: "God sent his Son, born of a woman [Mary, Mother of God], born under the law [Circumcision], ... that we might receive adoption as sons [Holy Name]"
In 2000, Pope St. John Paul II added an additional optional memorial of the Holy Name of Jesus on January 3, although it continues to be celebrated on Jan 1 (analogous to how Baptism of the Lord gets its own feast despite also being celebrated in the solemnity of the Epiphany).
