But the Queen Arwen said: 'A gift I will give you. For I am the daughter of Elrond. I shall not go with him now when he departs to the Havens; for mine is the choice of Lúthien, and as she so have I chosen, both the sweet and the bitter. But in my stead you shall go, Ring-bearer, when the time comes, and if you then desire it. If your hurts grieve you still and the memory of your burden is heavy, then you may pass into the West, until all your wounds and weariness are healed. 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 the chain about Frodo's neck. 'When the memory of the fear and the darkness troubles you,' she said, 'this will bring you aid.' (RotK, Book VI, Chapter VI, "Many Partings")

"In memory of Elfstone and Evenstar with whom your life has been woven!"

Indeed, Frodo's life was inextricably bound up in the fate of the Royal couple. Aragorn and Arwen had been charged not to wed until he became the King of both Gondor and Arnor, something that could not happen without the destruction of the One Ring. And Frodo, charged with seeing the Ring to its Doom, was the instrument through which the fate of Aragorn and Arwen's love could be achieved.

Aragorn's relationship with Frodo was one of guide and mentor, friend and comrade, companion on a desperate journey. He came to know Frodo as a friend, as indeed all the Company who left Rivendell together became bound in friendship.

Arwen did not know Frodo in the same way. She knew him only briefly before he left Rivendell, and through the reports to her of him by others. She very likely did come to consider him as a friend after getting to know him in Minas Tirith and on the journey to Edoras. There is no doubt that Arwen felt indebted to Frodo, who had secured her happiness for her, or that she felt compassion for his sufferings. These factors led her to offer to him her "white gem" and her "place" on the ship West. These story-internal factors are what drive her character to try to help the Ringbearer.

And yet there are other less obvious and more thematic story-external elements that tie the two characters together.

Choices

Each of them faced a choice, one which when taken, would sunder them from their own kin.

Arwen's choice was the choice faced by the peredhel, the choice her father and his brother had first faced, the choice her own brothers would eventually face: whether to follow the path of the Firstborn, accept her immortality and eventual sailing to the West, or to follow the path of the Secondborn, and accept the Gift of Ilúvatar and leave the Circles of Arda behind for an unknown fate.

"The Half-elven, such as Elrond and Arwen, can choose to which kind and fate they shall belong: choose once and for all. Hence the grief at the parting of Elrond and Arwen." (Letter #154)

It is fairly clearly implied that had it not been for her love for Aragorn, Arwen would have chosen to go with her father. "But there will be no choice before Arwen my beloved, unless you, Aragorn Arathorn's son, come between us and bring one of us, you or me, to a bitter parting beyond the end of the world. You do not know yet what you desire of me" is what Elrond says to Aragorn in App. A, v. which indicates that up until that point, Arwen thought she knew what her choice would be, and that it would be "no choice" -- in other words, to continue on as she had as one of the Eldar.

Therefore, when she admitted her love for Aragorn on Cerin Amroth, she had at that time made her choice to forsake the immortality of the Elves. She chose Lúthien's fate, which would end up causing her to be separated from her father and possibly from her brothers, and to forsake a reunion with her mother in the West.

Frodo's choice was different. Unlike Arwen, his choice was not a birthright, nor something he had known about all his life. And it would be, perhaps, more appropriate to call them choices, because while each choice he made led him on to the same end, he made the decision in stages: he chose to leave the Shire; he chose to take the Ring on the Quest; he chose to go on from Parth Galen alone; he chose with each step of the way to continue. And each of those choices was made when he decided to put the good of others ahead of his own.

Intertwining Fates

Frodo's accomplishment of the Ring's destruction (and Tolkien is clear that this was Frodo's doing, even if not in the way Frodo had envisioned) was what enabled Aragorn to claim the kingship, and thus, his bride. It is very clear that a "happy ending" for Aragorn and Arwen rested upon Frodo's small shoulders. So while his choices were not as clear-cut and singular as Arwen's, the way in which he impacted her life and destiny were very clear.

Arwen's wisdom and compassion enabled her to see first what others had yet to see clearly -- that Frodo was fading. One can speculate that perhaps it had been her experience with her mother's fading that enabled her to see the signs that those closer to Frodo (such as Aragorn and Gandalf) had failed to notice, or had perhaps refused to notice. We know that Frodo's passage to the West was her idea, as Tolkien mentions it more than once in his Letters:

He appears at first to have had no sense of guilt (III 224-5); he was restored to sanity and peace. But then he thought that he had given his life in sacrifice: he expected to die very soon. But he did not, and one can observe the disquiet growing in him. Arwen was the first to observe the signs, and gave him her jewel for comfort, and thought of a way of healing him.* Slowly he fades 'out of the picture', saying and doing less and less. I think it is clear on reflection to an attentive reader that when his dark times came upon him and he was conscious of being 'wounded by knife, sting and tooth and a long burden' (III 268) it was not only nightmare memories of past horrors that afflicted him, but also unreasoning self-reproach: he saw himself and all he had done as a broken failure."

* "It is not made explicit how she could arrange this. She could not of course just transfer her ticket on the boat like that! For any except those of the Elvish race 'sailing West' was not permitted, and any exception required 'authority', and she was not in direct communication with the Valar, especially not since her choice to become 'mortal'. What is meant is that it was Arwen who first thought of sending Frodo into the West, and put in a plea for him to Gandalf (direct or through Galadriel, or both), and she used her own renunciation of the right to go West as an argument. Her renunciation and suffering were related to and enmeshed with Frodo's: both were parts of a plan for the regeneration of the state of Men. Her prayer might therefore be especially effective, and her plan have a certain equity of exchange. No doubt it was Gandalf who was the authority that accepted her plea. The Appendices show clearly that he was an emissary of the Valar, and virtually their plenipotentiary in accomplishing the plan against Sauron. He was also in special accord with Círdan the Ship-master, who had surrendered to him his ring, and so placed himself under Gandalf's command. Since Gandalf himself went on the Ship there would be so to speak no trouble either at embarking or at the landing" (Letter #246)

We know that when Tolkien began writing LoTR, Arwen was not originally thought of as Aragorn's true love. But his love of parallelism and his need to bring to fruition the fate of Eärendil's line at the end of the Third Age led to her creation. Thematically she fits Aragorn far better than the original draft pairing of Aragorn and Éowyn, (just as thematically Faramir and Éowyn are a better pairing). She enables the lines of Men and Elves to be connected one more time, before the time of the Elves passes.

Mirrored Sacrifices

Both Frodo and Arwen made sacrifices.

Frodo sacrificed his innocence and life as he had always known it for the protection of the home he could no longer enjoy.

Arwen sacrificed an immortal life in Arda for an eternal love beyond the world.

The culmination of the sacrifice for both of them was the sailing of the grey ship at the Havens. The ship that was to have carried Arwen with her father to a reunion with her mother in Elvenhome instead carried Frodo Baggins away from the life he had hoped to have and those whom he loved, save Bilbo and Gandalf. He was leaving on the chance he would find healing, but it was a loss nonetheless.

Yet Frodo's journey to the West could not have been possible without the intervention and sacrifice of Arwen. Because she noticed that he was fading, she was able to offer the solution of sending him to the West -- to find the healing that she hoped her mother had found all those centuries before. In addition, this was a way of offering a partial payment of the debt she owed to Frodo in enabling her to attain her heart's desire of marriage to Aragorn.

Fleming Rutledge, in her book, The Battle for Middle-earth: Tolkien's Divine Design in The Lord of the Rings discusses how Arwen's gift was a significant exchange.

She reminds us that "The Lord of the Rings is not a 'happily-ever-after' story. It is a story of cross-bearing, and therefore a story of wounds, scars, and loss."

In a highly symbolic way, Arwen exchanges places with Frodo. Rutledge goes on to say:

Here then, is another Christ-like action by one of the personages in the Ring saga, one of the greatest of all the sacrificial actions in the story. Arwen, of course, did not make her sacrifice for Frodo; she did it for Aragorn and for their love. But this gift to Frodo bears with it the idea of exchange -- one life for the deliverance of another -- that is at the center of the Christian story.

Our impression of the new Queen's qualities of empathy and kindness are further confirmed when she also gives Frodo a diamond that is hanging on a chain about her neck, saying 'When the memory of the fear and the darkness troubles you, this will bring you aid'. She knows that the happiness of the moment cannot compensate for the pain he bears. It can't be emphasized enough --Tolkien at no point succumbs to the temptation to banish ambiguity and wrap everything up neatly. ( The Battle for Middle-earth: Tolkien's Divine Design in The Lord of the Rings, Fleming Rutledge, p. 356-7)

In many ways Frodo and Arwen were kindred spirits, united by their selflessness, their compassion, their wisdom and their mercy. It is only meet that in the end, Arwen celebrates the achievement of her own heart's desire by offering Frodo the possibility of his: healing and peace of spirit.

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Sources:

The Lord of the Rings, J.R.R. Tolkien
The Letters of J.R.R. Tolkien, Humphrey Carpenter, editor
The Battle for Middle-earth: Tolkien's Divine Design in The Lord of the Rings, Fleming Rutledge