The lectionary gets complicated when Christmas Day falls on a Sunday. Seven days after Christmas, January 1st, can be interpreted one of three ways:
January 1st can be "The First Sunday of Christmas" since it's the first Sunday after Christmas Day, and the gospel would be John 1:1-18.
January 1st can be "The Second Sunday of Christmas" since Christmas Day was also a Sunday, and the gospel would be Matt. 2:1-12.
January 1st can be "The Circumcision & Holy Name of Our Lord Jesus Christ," since it's the 8th day of Christmas, commemorating the day on which Jesus would have been circumcised and received his name, according to the law. The gospel for this would be Luke 2:15-21.
According to the Book of Common Prayer, "Circumcision & Holy Name" is a "Holy Day", for which these are the rules: "Any of these feasts that fall on a Sunday, other than in Advent, Lent, and Easter, may be observed on that Sunday or transferred to the nearest following weekday" (pg. 689, emphasis mine).
Given the permissive “may”, I'd choose Option 3: Circumcision & Holy Name. Why? For one, Holy Days are great, underappreciated, and worth celebrating. What seals it for me is that the gospel text (Luke 2:15-21) continues the narrative from the previous Sunday (Luke 2:1-14), making it easier to bury our lives in the Bible:
Luke 2:15-21
When the angels went away from them into heaven, the shepherds said to one another, “Let us go over to Bethlehem and see this thing that has happened, which the Lord has made known to us.” And they went with haste and found Mary and Joseph, and the baby lying in a manger. And when they saw it, they made known the saying that had been told them concerning this child. And all who heard it wondered at what the shepherds told them. But Mary treasured up all these things, pondering them in her heart. And the shepherds returned, glorifying and praising God for all they had heard and seen, as it had been told them.
And at the end of eight days, when he was circumcised, he was called Jesus, the name given by the angel before he was conceived in the womb.
It helps to know
The circumcision law. It's a single sentence: "On the eighth day the flesh of his foreskin shall be circumcised" (Lev. 12:3).
Its precedent. Leviticus simply encodes the words Yahweh spoke to Abraham in Gen. 17:1-14: "He who is eight days old among you shall be circumcised." In this text, Yahweh has just cut a covenant with Abram, now Abraham. He names him Abraham because he will be a "father of many nations." Circumcision, a cut in the male generative organ, is associated with fruitfulness. The old (his foreskin) is cut away, the new skin beneath emerges, and fruitfulness follows. The law ensures that every male among Israel participates in this meaningful ritual, including Jesus.
Why circumcision? There are medical reasons, cultic reasons, and comparative religion reasons. But there is also a biblical (and rabbinic) tradition that "Adam emerged already circumcised" (Avot D'Rabbi Natan 2.5). Circumcision restores a Son of Abraham to the dignity of Adam and calls him to the original Adamic vocation—fruitfulness—by revealing and making vulnerable the organ of fruitfulness.
Why eight days? Many reasons have been hypothesized. The most biblically-straightforward (also rabbinic) is this: Legislating an eighth-day circumcision ensures that every circumcisee will have participated in one Sabbath. Circumcision is "the sign" of the Abrahamic covenant, and the Sabbath is "the sign" of the Mosaic covenant.
In summary, an eighth day circumcision marks a male child as a participant in Adam, installed with the vocation of bearing fruit; in Abraham, part of the new-cut family of God; and in Moses, who received the law and the promise of new land.
What I noticed
The key verse is Luke 2:21. The previous paragraph, vv. 15-20, does not appear to be deeply related to it.
Some lectionaries take Luke 2:21-24 or Luke 2:21-40 as their gospel text. By doing so, they start with the Circumcision and the Name (v. 21), include the rest of the Presentation in the Temple (vv. 22-24), and reflect on his Reception by Simeon and Anna (vv. 25-38), and his Return to Nazareth (vv. 39-40).
Our (ACNA) lectionary gives us the name of the Holy Day—The Circumcision & Holy Name of Our Lord Jesus Christ—and then just one verse about those things (v. 21).
Although the Book of Common Prayer permits the clergy to extend the gospel reading, it seems appropriate not to extend this passage beyond the verse for which the Holy Day takes its name.
Grammatically, the clause about Jesus' circumcision is subordinated to the clause about his naming—"when he was circumcised, he was called Jesus"—and the naming is followed by the next, explanatory clause: "the name given by the angel before he was conceived in the womb." Luke appears to be more interested in the name than the circumcision.
Luke 2:21 is an obedience text. It simply repeats verbatim words that had been said earlier to demonstrate that the word of command was obeyed.
1:31 reads, "The angel said... you will conceive in your womb and bear a son, and you shall call his name Jesus." Then 2:21, "He was called Jesus, the name given by the angel before he was conceived in the womb."
There is nothing especially artful or eye-catching in the repetition of these words.
What I noticed this time was not anything about Jesus' name, but the significance of the announcement of his name.
Yahweh didn't share his name with Israel for a while.
Throughout Genesis, and most of Exodus, he is identified, lexically, as a species of the genus, "god" (elohim). There are many gods. He is one of them, albeit the only living one. Or, he is further specified by historical affiliation: "The God (elohim) of your fathers, of Abraham, of Isaac, and of Jacob."
My point is that "God" isn't a name; it's a term.
The God described in Scripture doesn't share his name until he shares it in Exodus 34, which, it is no coincidence, is today's Old Testament lesson:
The Lord descended in the cloud and stood with him there, and proclaimed the name of the Lord. The Lord passed before him and proclaimed, “Yahweh, Yahweh, a God merciful and gracious, slow to anger, and abounding in steadfast love and faithfulness, keeping steadfast love for thousands, forgiving iniquity and transgression and sin, but who will by no means clear the guilty, visiting the iniquity of the fathers on the children and the children's children, to the third and the fourth generation.” (vv. 5-7)
Yahweh shares his name because he had given it to Israel, and they now carried it. Taking Yahweh's name came with a specific responsibility, stated in the Decalogue: "You shall not take the name of Yahweh your God in vain, for the Lord will not hold him guiltless who takes his name in vain" (20:7).
When I was a kid, we were told taking God's name in vain meant saying, "Goddammit" instead of "Gosh darn it", or "Jesus!" instead of "Jeez!"
But that's not what Yahweh means. He means that, now that he has given this people his name, they can't live as if they don't belong to him. (Carmen Imes recently wrote a dissertation showing this; here's her book).
Israel may have had their own "Feast of the Holy Name of Yahweh". And that would have been good. Reflecting on the Revelation, the Meaning, and the Giving of the name of their God would have helped them keep the commandment. A rich faith community knows how to pair holidays with their principles or laws.
We do have a "Feast of the Holy Name of Jesus Christ." He is one with with the Father, but he has his own name. And the angel gives an announcement that's not unlike Yahweh's in Exodus 34:
"...You shall call his name Jesus. He will be great and will be called the Son of the Most High. And the Lord God will give to him the throne of his father David, and he will reign over the house of Jacob forever, and of his kingdom there will be no end." (Luke 1:31-33)
Luke tells of the Revelation, the Meaning, and the Giving of the name of our God, Jesus Christ. The commandment in Exodus still applies to us: "You shall not take [on] the name of Jesus Christ in vain."
The name of Jesus was given to us. We have taken it. Praise God, and may we not take it in vain!
Lectionary Resonances
Exodus 34:1-9 is the announcement of Yahweh's name, and the OT parallel to Luke 2:21. Everything Yahweh says about himself is true of Jesus.
In seven verses, St. Paul writes the name "Jesus Christ" four times (Rom. 1:1-7). But look at the way he talks about Jesus in vv. 5-6: Jesus Christ, "through whom we have received grace and apostleship to bring about the obedience of faith for the sake of his name among all the nations, including you who are called to belong to Jesus Christ...". This part of Paul’s ministry can be understood as the conclusion of two biblical premises: Our God says not to take his name in vain (Exod. 20:7) and our God’s name is Jesus (Luke 2:21). The office Paul has received is for the sake of bringing about obedience among those who belong to Jesus. That obedience has to do with Jesus' name throughout the world.
Paul's concern has to do with Psalm 8. Verses 1 and 9 of Psalm 8 repeat, forming an inclusio around the other seven verses: "O Yahweh, our Lord, how majestic is your name in all the earth!" The purpose of the existence of Paul's apostolic office, and the existence of all the institutions of the Church—the Scripture, the lectionary, the liturgy, the sacraments, the clergy, the calendar, etc.—is to bring about the obedience that majestifies the name of Jesus. "What is man that you are mindful of him?" (v. 4) Man is called to take Jesus' name, to obey him, and bear fruit.
I'll close by claiming, candidly, and citing no historical evidence, that the Western Church knew what she was doing when she put these twin feasts eight days after Christmas, on the Roman calendar's New Year's Day.
In the Church we don’t conceive of "New Year’s Resolutions" as self-determined goals for self-improvement.
"Resolutions", described by these texts, must simply be the obedience for which the institutions of the Church exist, and which majestify the name of Jesus.