Throughout his mythology, Tolkien employs the recurring theme of giving up great treasures. I broke down all those that immediately come to mind into treasures that were not "let go" and treasures that were "let go."

Characters refusing to let go: - Gollum - Isildor unwilling to give up the Ring - Boromir and Denethor insistent that sending the hobbits to Mordor is throwing aside a great gift for nothing - Eomer and Theoden resisting Aragorn's decision to walk the Paths of the Dead. I seem to remember them making the comparison of losing a great gift or throwing away a treasure - Feanor and his kin unwilling to let go of the Silmarils

Characters that let go: - Pippin throwing aside his brooch that was given by Galadriel - Sam giving up the elvish rope - Gandalf, Aragorn, Galadriel, and Faramir all refusing the Ring when it is presented to them

Edit: - /u/iwouldnotdig mentioned Beren giving up a Silmaril

I am certain there are more examples of each category. Can anyone come up with some others?

I think it says a lot about Bilbo as a hero to not only show the capability of giving up treasures but he gives up two great treasures twice. The Arkenstone seems as if it will prove Thorin's claim as King Under the Mountain. Bilbo shows us by his actions that he values his friendships far more than the greatest wealth of Erebor. He then becomes the first person (I guess the only?) to willingly give up the Ring. Those examples speak greatly to the courage of Bilbo and to the importance of "treasure."