The name Lucifer does not appear in the Hebrew text. It seems that it was added or inserted in the Latin translation as a replacement for "morning star".
Excerpt from A Pilgrim's Path by John J. Robinson:
". In the original Hebrew text, the fourteenth chapter of Isaiah is not about a fallen angel, but about a fallen Babylonian king, who during his lifetime had persecuted the children of Israel. It contains no mention of Satan, either by name or reference. The Hebrew scholar could only speculate that some early Christian scribes, writing in the Latin tongue used by the Church, had decided for themselves that they wanted the story to be about a fallen angel, a creature not even mentioned in the original Hebrew text, and to whom they gave the name "Lucifer."
Why Lucifer? In Roman astronomy, Lucifer was the name given to the morning star (the star we now know by another Roman name, Venus). The morning star appears in the heavens just before dawn, heralding the rising sun. The name derives from the Latin term lucem ferre, bringer, or bearer, of light." In the Hebrew text the expression used to describe the Babylonian king before his death is Helal, son of Shahar, which can best be translated as "Day star, son of the Dawn." The name evokes the golden glitter of a proud king's dress and court (much as his personal splendor earned for King Louis XIV of France the appellation, "The Sun King").
The scholars authorized by ... King James I to translate the Bible into current English did not use the original Hebrew texts, but used versions translated ... largely by St. Jerome in the fourth century. Jerome had mistranslated the Hebraic metaphor, "Day star, son of the Dawn," as "Lucifer," and over the centuries a metamorphosis took place. Lucifer the morning star became a disobedient angel, cast out of heaven to rule eternally in hell. Theologians, writers, and poets interwove the myth with the doctrine of the Fall, and in Christian tradition Lucifer is now the same as Satan, the Devil, and --- ironically --- the Prince of Darkness.
So "Lucifer" is nothing more than an ancient Latin name for the morning star, the bringer of light. That can be confusing for Christians who identify Christ himself as the morning star, a term used as a central theme in many Christian sermons. Jesus refers to himself as the morning star in Revelation 22:16: "I Jesus have sent mine angel to testify unto you these things in the churches. I am the root and the offspring of David, and the bright and morning star." Source: here
Further, "son of the morning" may very well have been an ironic appellation by God by applying the meaning of a pagan god's name to the king of Babylon.
As answered in a previous question here, the Babylonian pagan god Attar was Venus, which is the morning star. And, as many ancient kings claimed divine right of rule by appropriating pagan gods for their sponsor or parent, they often claimed to be the son of whichever pagan god told them they had the right to rule.
So, "son of the morning" most probably was God's way of calling out the king of Babylon in righteous indignation for reaching too high.