Why do the Dunedain wear Fëanorian stars?

Question

Why do the Dunedain wear Fëanorian stars? Is it because of their Noldorin descent through Ëarendil?

Answer 1

This is the "Star of the Dúnedain", not a Fëanorian star. We do not know its origin or significance.

You are probably referring to the star that can be seen in the following two passages from The Return of the King describing rangers.

A little apart the Rangers sat, silent, in an ordered company, armed with spear and bow and sword. They were clad in cloaks of dark grey, and their hoods were cast now over helm and head. Their horses were strong and of proud bearing, but rough-haired; and one stood there without a rider, Aragorn’s own horse that they had brought from the North; Roheryn was his name. There was no gleam of stone or gold, nor any fair thing in all their gear and harness: nor did their riders bear any badge or token, save only that each cloak was pinned upon the left shoulder by a brooch of silver shaped like a rayed star.
_{The Lord of the Rings - Book V, Chapter 2 - "The Passing of the Grey Company"}

[Ecthelion II] had the aid and advice of a great captain whom he loved above all. Thorongil men called him in Gondor, the Eagle of the Star, for he was swift and keen-eyed, and wore a silver star upon his cloak; but no one knew his true name nor in what land he was born.
_{The Lord of the Rings - Appendix A - "Annals of the Kings and Rulers"}

This star is mentioned by name in Appendix B

1436 King Elessar rides north, and dwells for a while by Lake Evendim. He comes to the Brandywine Bridge, and there greets his friends. He gives the Star of the Dúnedain to Master Samwise, and Elanor is made a maid of honour to Queen Arwen.
_{The Lord of the Rings - Appendix B - "The Tale of Years"}

The connection between the two is not obvious. Robert Foster missed it when writing his The Complete Guide to Middle-earth (instead attributing the Star of the Dúnedain to the Elendilmir), and in Unfinished Tales, Christopher just noted that Foster's assumption was wrong, but that he himself wasn't sure what the Star of the Dúnedain was either. Later in The Treason of Isengard, Christopher wrote that readers had made the connection to the rangers' brooch and that this was "clearly correct".

This is a convenient place to mention that after the publication of Unfinished Tales two correspondents, Major Stephen M. Lott and Mrs. Joy Mercer, independently suggested to me that the Star of the Dunedain was very probably the same as the silver brooch shaped like a rayed star that was worn by the Rangers ... These suggestions are clearly correct.
_{The Treason of Isengard - Part Three, Chapter IV - "Many Roads Lead Eastward (1) - Note 8}

Nothing else is known about this star though, and we do not know its origin or its purpose.