The traditional Roman calendar associates this day with the Holy Name of Jesus. It used to be associated with the day before, with the Feast of the Circumcision. (In fact, the Gospel reading for both feasts is identical.) Then in 1913, Pope Pius X moved it to the Sunday between the second and the fifth January inclusive, and in years when no such Sunday existed, to be observed on the second of January. Don't ask me why.
Historically, the observance of this feast has been all over the place until nearly one hundred years ago. The circumcision of a newborn male under Jewish law must take place eight days after the child's birth, at which time he is given his name. Small wonder, then, that the Gospel readings for both feasts in the traditional Roman calendar are the same. Some Western traditions, such as Anglican and Lutheran, celebrate both on the first of January, as did the Roman for quite some time -- you know, being the eighth day and all.
And speaking of names ...
I once heard a comedian pose this important theological question: “If Jesus was Jewish, why did He have an Hispanic name?” That occasion aside, it gives us an occasion of our own, to consider that the name "Jesus" was not an uncommon one in His day. Brian Palmer writes for Slate:
Many people shared the name. Christ's given name, commonly Romanized as Yeshua, was quite common in first-century Galilee. (Jesus comes from the transliteration of Yeshua into Greek and then English.) Archaeologists have unearthed the tombs of 71 Yeshuas from the period of Jesus' death. The name also appears 30 times in the Old Testament in reference to four separate characters -- including a descendant of Aaron who helped to distribute offerings of grain (2 Chronicles 31:15) and a man who accompanied former captives of Nebuchadnezzar back to Jerusalem (Ezra 2:2) ...
How would Christ have been addressed by those around him?
Certainly not as "Mister Christ." In fact, "Christ" was not a name, but an honorific, a title if you will, from the Greek Khristós for "anointed one." The Hebrew word was Moshiach or "Messiah." He would have been known by His given name, and the name of His father -- “Yeshua bar Yehosef” or “Jesus Son of Joseph.” In later centuries (or in present-day Iceland), we might easily surmise His having been addressed as “Jesus Josephson.”
We also know that He eventually left Nazareth in Galilee, the town of His childhood, for other parts of that country, as well as Samaria and Judea. In those places, He would have been just as likely addressed as “Yeshua Nasraya” or “Jesus of Nazareth.” We know this from Scripture, as this was the inscription on the Cross, which gave both His name and His offense, in three languages: “Jesus of Nazareth, King of the Jews” (actually, “Iesus Nazarenus Rex Iudaeorum” in Latin, “Ihsoûs ó Nazoraîos ó Basileùs tôn ’Ioudaìov” in Greek, and “Yeshua HaNazarei v Melech HaYehudim” in Hebrew). After all, a guy from a hick town like Nazareth would have been rather conspicuous in a high-falutin' place like Jerusalem, especially outside of the High Holydays.
The Scriptures also record him being addressed as “Jesus Son of David.” A man would also have been known for his extended family; that is, his tribe or house, as in “Yeshua ben David” or “Jesus of the House of David.” Or so I've read. But even though family lineage was everything in Jewish society, such an address was not as common in everyday use.
Or so I've read.
Devotion to the Holy Name has also been the inspiration for the National Association of the Holy Name Society. HNS chapters have been the basis for men's clubs in Catholic parishes for generations.