The Star of the Dúnedain was the badge of kinship among the Dúnedain of Arnor, the Rangers of the North.

As recorded in LotR:
"...there was no gleam or stone or gold, nor of any fair thing in all their harness and gear; nor did the riders [Rangers] bear any token or badge, save only that each cloak was pinned on the left shoulder by a brooch of silver shaped like a rayed star." (Book V, Passing of the Grey Company, RotK).

The Star of the Dúnedain was worn with pride and honor, and seldom bestowed upon others.

It is said in the Tale of Years that in SR 1436, upon visiting his northern realm, the King Elessar accorded Samwise Gamgee, Mayor of the Shire, a Star of the Dúnedain, this being the only recorded instance of the Star being given to one not of Númenoreans.

**Note: In some extant writings of Tolkien's, the name "Star of the Dúnedain" also referred to the Elendilmir, the sign of royalty of the Northern Line. But, this jewel was more correctly referred to as the Star of the North (as being unique to Arnor), or the Star of Elendil.

Since the Elendilmir was so greatly prized and revered by the descendants of Elendil for its ancientry and beauty, it is hardly conceivable that they would give it away lightly, even to someone as esteemed as Samwise Gamgee.

Suffice it to say, the Star of the Dúnedain, most likely refers to just that, and not the Elendilmir at all.
Encyclopedia entry originally written by Nolimon