Underlying these words is profound connection between these two characters. As Tolkien notes in Letter 246, Arwen was the first to note the signs of the disquiet growing in Frodo and she "game him her jewel for comfort, and thought of a way of healing him." He then added in footnote: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."
Indeed, I would go so far as to say that the special relationship between Frodo and Arwen goes to the very heart of what Tolkien repeatedly called the "real theme" of LOTR: Death and Immortality. After all, it is primarily through these two characters that Tolkien illustrates this theme, even if much of Arwen's part of the story is moved to the crucial appendix. I find the idea that her renunciation and suffering were related to Frodo's a fascinating one. It is quite obvious how her renunciation and suffering were part of the 'plan' for the regeneration of men, and I guess it is equally true that without the success of Frodo's quest (despite his own personal 'failure') that regeneration would have been impossible.What is meant is that is 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 specially effective, and her plan have a certain equity of exchange.
Yes, there was a special bond between these two disparate characters.