Arwen Undomiel (or Evenstar) is called this because she is the last of the beautiful of her elven race. It is sort of ironic then that she leaves behind her elven heritage to embrace the small part of her that is mortal.
In a way, you can say that Arwen's life is a reflection of the elves tenure in the lands of Middle Earth.
The name Undomiel is derived from the elven word Undome, which means Twilight, which can also mean evening. Undomiel translates to mean roughly "Twilight Maiden" or "Evening Maiden". Then in the common tongue, this phrase has been twisted a bit to mean "Evenstar" or Evening Star.
Arwen was given the name Undomiel by her grandmother Galadriel. Galadriel saw in her granddaughter the last of a long line of Elves to inhabit the lands of Middle Earth. She saw her as the twilight of the Elven race.
Arwen Evenstar goes on to prove her namesake as being correct. She is not only the last of the Elves, she actively chooses to no longer be an elf when she decides to give up her immortality for Aragorn. This can be compared to the Elven race itself, which actively chooses to leave the lands of Middle Earth to go to the Undying Lands. Both Arwen Evenstar and the Elves give up one kind of immortality to choose another (Arwen's is the immortality of her love while the Elves' is the new immortality of the Undying Lands).
Arwen Evenstar is considered to be the most beautiful of her race and she comes at the end of it. Much in the same way the most illustrious involvement if the Elves in Middle Earth comes right at the end of their tenure in Middle Earth.
Both Arwen Evenstar and the Elves sacrifice much for man. Arwen's sacrifice is for one man and the Elves sacrifices come for the race of Men. The Elves leave Middle Earth so that Men can rule there as their time has come. Arwen leaves behind the Elves so that her love, Aragorn can rule the race of Men.
Galadriel is shown several times throughout the Tolkien writings as having a kind of second sight. It is then not too far of a reach to say that her bestowing the name Undomiel or Evenstar on Arwen was one of fancy but rather a knowledgeable hint as to what the future of the Elves would look like.