What is the name of the star that shines as the morning star in Middle-earth?

The star that shines as the morning star in Middle-earth is Eärendil's star. In Tolkien's mythology, Eärendil the Mariner sails into the West carrying a Silmaril, a jewel containing light from the Two Trees of Valinor. The Valar place him in the sky, where he becomes a beacon of hope for the peoples of Middle-earth.

The star is associated with Venus, which can appear brilliantly before sunrise as the morning star. Tolkien drew inspiration from the Old English name Ēarendel and from the poem Crist, whose “brightest of angels” imagery helped shape his mythology. The name Eärendil itself belongs to Tolkien's legendarium, while its linguistic roots reach into older Germanic and Anglo-Saxon traditions.

A common mix-up is confusing Eärendil's star with the Star of Elendil or with the Phial of Galadriel. The Phial is a small vessel containing some of the light of Eärendil's star; it is not the star itself. Frodo invokes that light with the Quenya phrase “Aiya Eärendil Elenion Ancalima!” in The Two Towers.