Q:

Why is the light of Eärendil the Elves' most beloved star?

Wasn't Eärendil a mortal man? So why would he be most beloved to Elves?

A:

Earendil was half-Elven, who chose to be an Elf.
Why would his star have been particularly beloved?

It was actually one of the Silmarils, the last true light of the Two Trees of Valinor, which the Noldor had fought and died to recover.
It was a sign from the Valar: after hundreds of years of despair and defeat against Morgoth, the Star of Earendil was a symbol that the Valar (and Eru) were still with them, and a reminder of the War of Wrath in which Morgoth was utterly defeated and the Elves freed from his domination of Middle-Earth.

