For those of us who live the Church's calendar - by saying the Divine Office, by attending Mass through the week - the joy of a great celebration can seem a little flattened by the ordinariness of the days after: back to the 'ordinary'; back to green vestments. This is why liturgical traditions have tended to give us a gentle let-down. In the eastern traditions, very often the 'day after' is an associated celebration; a rendering of the same theme in a different key. Thus the Byzantines and the East and West Syrians keep December 26 as the commemoration of Mary, the Mother of God. In our western traditions, the Octave has performed a similar role. For a week after the Great Day the festival continues to colour our worship. Then, on the eighth day, which will be the same day of the week, we say farewell to the festival by celebrating it again ...but, of course, at a reduced level.
In my view, it was unfortunate that the 'reforms' of the 1960s almost entirely eliminated the concept of the Octave. Out of the window went the practice of keeping a Sunday as the 'Sunday within the octave of-such-and-such'. We now have only the octaves of Christmass and Easter. Although: the discerning eye can see one or two shadows of the old Octave Days; the eighth day after the Assumption, August 22 is the Feast of our Lady, Queen; and in the Church of England November 8 is the Feast of All Saints of England. The old Octave Day of our Lady's Birthday, September 15 - also the day after Holy Cross Day - is, very neatly, the memorial of our Lady at the Cross.
The Octave of Christmass, January 1, has long been marked in the West by texts which take the Divine Maternity of Mary as their theme. She is celebrated as Theotokos: Mother of God. This is a safeguard of the Divinity of our Lord; he is God and so his mother is Mother of God. (Sometimes you hear it suggested that Godbirthgiver would be a better translation, as if birthgiver is a more natural word than mother. The Orthodox certainly don't think so: at the top of every ikon of our Lady are the words Meter Theou: 'Mother of God' quite literally.)
Jesus was circumcised on the eighth day, and January 1 came to be called in the Middle Ages the Feast of the Circumcision, although the Lord's circumcision was barely mentioned in the texts used until Cranmer got to work. The revised Roman rite wisely calls the day 'Mary Mother of God' because that is what the ancient texts are all about. Possibly because of squeamishness, ASB and CW restyled the day 'Name of Jesus', since we are told that is when he was given his name. And Rome has recently placed that commemoration on January 3 (personally I rather like the Sarum and BCP notion of celebrating the Holy Name on August 7, the day after the Transfiguration).