Tolkien only hints at her role when she first appears: even Frodo is smitten. There are tiny clues about what she means to Aragorn there. Bilbo's first comment to him is, "why weren't you at the feast? Lady Arwen was there!" letting us know she matters a great deal to Aragorn. Aragorn then insists to Bilbo that he include a green stone in the Lay of Eärendil, namely, the Elessar of Idril.
Immediately after that we see Aragorn visibly transformed into a prophetic vision of what he will become, something which happens repeatedly in the book:
Near him sat the Lady Arwen. To his surprise Frodo saw that Aragorn stood beside her; his dark cloak was thrown back, and he seemed to be clad in elven-mail, and a star shone on his breast.
Two things come out of this image: the idea of kingly-Aragorn appearing when he is with Arwen, and the star on his breast. Not the Elessar, but a harbinger of it.
In Lórien we see this sort of transformation happen again:
For the grim years were removed from the face of Aragorn, and he
seemed clothed in white, a young lord tall and fair; and he spoke words in the Elvish
tongue to one whom Frodo could not see. Arwen vanimelda, namárië!
he said, and then he drew a breath, and returning out of his thought he looked at
Frodo
and smiled.
`Here is the heart of Elvendom on earth,' he said, `and here my heart dwells ever,
unless there be a light beyond the dark roads that we still must tread, you and I. Come
with me! ' And taking Frodo's hand in his, he left the hill of Cerin Amroth and came there
never again as living man.
Again, Aragorn is transformed into a lordly figure in proximity (physical although not temporal) with Arwen, and his greatness shines through. His words forcefully acknowledge how closely his own path and Arwen's are linked, and the memory which he is reliving is a very specific one, not explained until the appendix: the moment when he and Arwen pledged to fight the Shadow together. She doesn't fight at his side like Lúthien, but there could be no clearer way (save Jackson's) to show that, in Aragorn's mind, she is with him on his journey. Also this moment predicts their eventual reunion in Cerin Amroth after he ceases to be a living man, i.e., when she dies on Cerin Amroth (we hope, from these enigmatic words) their spirits will meet here to take their last journey.
The Cerin Amroth scene is full of mystical import for the character of Aragorn and has mythical echoes of Avalon, where the priestess anoints the king. There is little reason for this scene to be here, for the party to stop at Cerin Amroth before going on, except us to remind us of Aragorn's strong connection of the heart to Arwen, and to show what makes the king of men tick. It is a striking moment.
Finally, of course, Arwen's gift is literally his royal name... she is not there, yet still, she "anoints" him, and again we see the prophetic, visual transformation of Aragorn which heralds his future greatness:
Then Aragorn took the stone and pinned the brooch upon his breast, and those who saw him wondered; for they had not marked before how tall and kingly he stood, and it seemed to them that many years of toil had fallen from his shoulders. `For the gifts that you have given me I thank you,' he said, 'O Lady of Ló rien of whom were sprung Celebrían and Arwen Evenstar. What praise could I say more? '
One could simply take his pretty words about the woman he loves to be, "Arwen's a love
interest," and leave it at that, but the moments when she's mentioned and the context are
always affirmations of Aragorn's kingship, the moments when the Elves tell him,
"yes, you will be king, you are hope." He is inspired and transformed by
these moments. He has just gotten a sheath for his sword and the healing stone that will
represent his more merciful aspect as king-- symbols of royalty, and one might be
tempted to read the old symbolism of a sword's sheath, if you know the Latin!
The next mention of Arwen is the moment when Aragorn stops following Gandalf's
counsel or the needs of the Fellowship and steps into his role as king and savior...
'Always my days have seemed to me too short to achieve my desire,' answered Aragorn.
'But great indeed will be my haste ere I take that road.'
'That will soon be seen,' said Elrohir. 'But let us speak no more of these things upon the
open road!'
And Aragorn said to Halbarad: 'What is that that you bear, kinsman?' For he saw that
instead of a spear he bore a tall staff, as it were a standard, but it was close-furled in a
black cloth bound about with many thongs.
'It is a gift that I bring you from the Lady of Rivendell,' answered Halbarad. 'She
wrought it in secret, and long was the making. But she also sends word to you: "The days
now are short. Either our hope cometh, or all hopes end. Therefore I send thee what I have
made for thee. Fare well, Elfstone!"'
And Aragorn said: 'Now I know what you bear. Bear it still for me a while!' And he
turned and looked away to the North under the great stars, and then he fell silent and
spoke no more while the night's journey lasted.
His desire? Does he merely mean the throne? I think not, and his thoughts of Arwen a moment later seem to bear me out: the end, the desire he is working for, is more than just the kingship and salvation of Middle-earth. I think he'd use another expression than "desire". Obviously he is yearning, and his years of effort, have been inspired by the challenge Elrond has set him, to be worthy of Arwen. It is certainly not the only reason, but here it is presented as a powerful motive, powerful enough to leave him speechless the rest of the night long. And her own words to him here demonstrate her role: she reminds him of who he is, and she keeps faith in him.
Again, she is anointing him as king in the most powerful of ways, sending him a banner-- a lady's favor, as the Elfstone was. The symbols and tokens she gives him are not just passive signs of affection or regard in this story; they are visible tools by which he rallies the hearts of men and convinces them he is the king! Her faith in him is channelled in such a way that others "catch the spark", as Frodo did when she first saw Arwen and Aragorn together-- they "grok" that Aragorn is special partly because of her. And recall that Tolkien defines Estel specifically as the sort of hope which is synonymous with trust, faith, an inner purpose or sense of direction not based on precedents or experience. She nurtures and affirms Aragorn's estel, which is the virtue by which he wins when Denethor and many other can't.
Arwen again shows up in name in the very moment of Aragorn's triumph, the exact
moment when "the king returns", and her agency in that moment is duly acknowledged:
And then wonder took him, and a great joy; and he cast his
sword up in the sunlight and sang as he caught it. And all eyes followed his gaze, and
behold! upon the foremost ship a great standard broke, and the wind displayed it as she
turned towards the Harlond. There flowered a White Tree, and that was for Gondor; but
Seven Stars were about it, and a high crown above it, the signs of Elendil that no lord had
borne for years beyond count. And the stars flamed in the sunlight, for they were wrought
of gems by Arwen daughter of Elrond; and the crown was bright in the morning, for it was
wrought of mithril and gold.
Thus came Aragorn son of Arathorn, Elessar, Isildur's heir, out of the Paths of the Dead,
borne upon a wind from the Sea to the kingdom of Gondor...
There are echoes of the story of the return of Elendil's heirs to Middle-earth in that, and Arwen's name is again forcefully connected with legitimizing and supporting Aragorn.
Last of all, Aragorn is not convinced of his renewal of the king's line and Gondor until he
receives the sign of his union with her, which is simultaneously the sign of the life
of Gondor itself-- the White Tree. Then, he knows he will marry her, and the line of Kings
will be renewed.
In short, Arwen is a personal inspiration to Aragorn's spirit, and she is the passive
kingmaker of the tale, as Gandalf is the active one.