"Elendilmir" was a name of the Star of Elendil, or Star of the North Kingdom, a jewel made of mithril. It became a token of royalty throughout the Kingdom of Arnor, having been first crafted and presented to the High King of Gondor and Arnor, Elendil the Tall, in the Second Age.
Elendilmir was lost in the Disaster of the Gladden Fields, when King Elendil's eldest son, Isildur, and the Dúnedain of Arnor were marching home after the defeat of Sauron. Only three survived, excluding Isildur; Elendilmir would have been lost with him in the Gladden River.
In Unfinished Tales (published by Christopher Tolkien awhile after The Lord of the Rings) it is stated that Aragorn II finds the Elendilmir in the tower of Orthanc. After Sauron's defeat, Aragorn (with the help of Gimli the Dwarf) finds a hidden door and a steel cabinet, in which they find a small golden tube without any inscription, containing the Elendilmir. Aragorn brings it back to Minas Tirith and wears it once again, but only on special occasions, to risk it no harm.
Another version of this gemstone was made at a later date by Isildur's descendants, a second Elendilmir, commemorating King Elendil and Isildur himself. It was borne roughly around TA 2940, or most likely later.