Arwen was the Evenstar of her people, the last great light of the race of Elves
So it was that Frodo saw her whom few mortals had yet seen; Arwen, daughter of Elrond, in whom it was said that the likeness of Lúthien had come on earth again; and she was called Undómiel, for she was the Evenstar of her people.
The Fellowship of the Ring - Book II, Chapter 1: Many Meetings
The tale of Arwen parallels that of Lúthien, in fact, Arwen was considered as beautiful as her foremother, who was considered the most beautiful elf of all times. The likeness of the two is noted by Aragorn, in the Tale of Aragorn and Arwen1
and his daughter Arwen Undómiel, who resembles Lúthien closely in looks and fate...
The Letters of J.R.R. Tolkien - Letter 141, To Naomi Mitchison
Arwen is not a 're-incarnation' of Lúthien (that in the view of this mythical history would be impossible, since Lúthien has died like a mortal and left the world of time) but a descendant very like her in looks, character, and fate.
The Letters of J.R.R. Tolkien - Letter 153, To Peter Hastings
for Lúthien was the most beautiful of all the Children of Ilúvatar.
The Silmarillion - Chapter 19: Of Beren and Luthien
The name Undomiel was given to Arwen to reflect her beauty during the time of the waning of the Elves in the Age. Arwen was a reflection of the ancient glory of the race of Elves in the Years of the Trees and the First Age at a time when the Firstborn had begun to fade, and the time of Men had begun. Arwen was part of the sixth, and final, union of the Children of Ilúvatar (after Luthien & Beren, Idril & Tuor, Nimloth & Dior, Elwing & Eärendil and Mithrellas and Imrazór). In plighting her troth with Aragorn, Arwen renews the long forgotten alliance between the two races of the Children. Their betrothal represents the passing of the torch, from the Firstborn who had ruled Middle-earth for millennia to the Second-born, who's time was rising. Furthermore, Arwen, alone of the High Elves remains in Middle-earth.
"But first I will plead this excuse," said Éomer. "Had I seen her in other company, I would have said all that you could wish. But now I will put Queen Arwen Evenstar first, and I am ready to do battle on my own part with any who deny me. Shall I call for my sword?"
Then Gimli bowed low. "Nay, you are excused for my part, lord," he said. "You have chosen the Evening; but my love is given to the Morning. And my heart forebodes that soon it will pass away for ever."
The Return of the King: Book VI, Chapter 6: Many Partings
The Evenstar remains in Middle-earth for a time after the beginning of the Age of Men, however the Morning (Galadriel, the last memory of a long forgotten age) passes away forever over the Sea. This again reflects Arwen's fate as the last shining light for the Firstborn in a world of Men.
1 For Aragorn had been singing a part of the Lay of Lúthien which tells of the meeting of Lúthien and Beren in the forest of Neldoreth. And behold! there Lúthien walked before his eyes in Rivendell, clad in a mantle of silver and blue, fair as the twilight in Elven-home; her dark hair strayed in a sudden wind, and her brows were bound with gems like stars.
For a moment Aragorn gazed in silence, but fearing that she would pass away and never be seen again, he called to her crying, Tinúviel, Tinúviel! even as Beren had done in the Elder Days long ago.
Then the maiden turned to him and smiled, and she said: “Who are you? And why do you call me by that name?”
And he answered: “Because I believed you to be indeed Lúthien Tinúviel, of whom I was singing. But if you are not she, then you walk in her likeness.”
“So many have said,” she answered gravely. “Yet her name is not mine. Though maybe my doom will be not unlike hers. But who are you?”
The Return of the King: Book VII, Appendix A, Part V: Here follows a part of the Tale of Aragorn and Arwen