Tolkien scholar Michael Martinez says that the Star was a symbol of law enforcement authority. He states:
"The badge worn by the Rangers may thus be designated a special royal corps, perhaps like a bodyguard in some people's estimation but more likely (in my opinion) as a body of officers bearing the King's authority.
This royal designation would have afforded the bearers of these badges special authority. In fact, in the entry for Shire year 1434, Tolkien writes:
"Peregrin becomes the Took and Thain. King Elessar makes the Thain, the Master and the Mayor counsellors of the North Kingdom. Master Samwise is elected Mayor for the second time".
It is notable that only Samwise receives the Star of the Dunedain but in his capacity as Mayor it is Sam who oversees the Shire's police forces: the Shirriffs and the Bounders. I think Tolkien meant Aragorn's "gift" to actually be a formal investiture of police authority in the Mayor of the Shire, so that the Shirriffs and the Bounders, in effect, became "Officers of the Realm" through the authority of the Mayoralty."
As the Fourth Age turned after the War of the Ring, we know that the Rangers of the North stayed near to Aragorn / King Elessar whilst the Rangers of Ithilien stayed near to Prince Faramir in the resettled land of Ithilien. All were Rangers and under the King and Prince's authority. More than likely this badge of authority was extended to the Rangers of Ithilien as well although their badge probably depicted the Tree of Gondor and Seven Stars.
After the War of the Ring, King Elessar visited with Samwise Gamgee, who by that time had become Mayor of the Shire. King Elessar presented Sam with the Star of the Dunedain.
Christopher Tolkien initially expressed ignorance of what his father intended (in Unfinished Tales of Numenor and Middle Earth) but in The War of the Ring, Vol. VIII of the History of Middle Earth, Christopher Tolkien added this note to a passage describing the Rangers of Eriador:
"8. In the Tale of Years (LR Appendix B), the entry for the year 1436 in the Shire Reckoning states that the King Elessar, coming to the Brandywine Bridge gave the Star of the Dunedain to Master Samwise. In my note 33 to The Disaster of the Gladden Fields in Unfinished Tales (pp. 284-5), I said I was unable to say what this was. This is a convenient place to mention that after the publication of Unfinished Tales, two corrospondents, 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 in the present passage (RK p. 51); Mrs. Mercer also referred to the star worn by Aragorn when he served in Gondor, as described in Appendix A (I, IV, The Stewards):
"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." These suggestions are clearly correct."
Here is an illustration by noted Polish artist Catherine Karina Chmiel - Gugulska that depicts Aragorn as Thorongil with Ecthelion and a young Boromir. Note the Star upon his cloak has five points.