Symbolism, Imagery, Allegory
When Arwen settles down with Aragorn as Queen of Gondor, she gives Frodo a white gem (yet again with the star imagery; see "The Seven Stars" for more on this significance). This gem is like the Phial of Galadriel but less powerful. It glows with comfort rather than light, and Frodo clutches it whenever his dreams and memories of Mordor get too pressing.
Arwen's gem hangs around Frodo's neck where the Ring used to hang, as though Arwen is trying physically to replace the Ring with her more positive star gem. She tells Frodo:
"But wear this now in memory of Elfstone and Evenstar with whom your life has been woven!" And she took a white gem like a star that lay upon her breast hanging upon a silver chain, and she set a chain about Frodo's neck. "When the memory of the fear and the darkness troubles you," she said, "this will bring you aid." (6.6.8)
Beyond giving Frodo comfort, Arwen's gem has a more immediate significance. It is proof of Arwen's offer to Frodo to take her place on the elf ships sailing away from Middle-earth to the West. So, the gem is not only reassuring; it is also the shiniest, most valuable boat ticket you ever saw.