This design has the Eye of Sauron large and on top, with searching rays emanating out, searching for the One Ring. Between the Eye and the Ring is the Elfstone, with wings stretched out blocking all view of the Ring. Aragorn is represented by the Elfstone here because the jewel is symbolic of him stepping into his role as king, and as the heir of Isildur he also had the right to look into the Palantir and challenge Sauron. Aragorn's proposal in this chapter is to draw Sauron's attention to himself, providing protection to Frodo by keeping him secret.
“And word went through the City: 'The King is come again indeed.' And they named him Elfstone, because of the green stone that he wore, and so the name which is was foretold at his birth that he should bear was chosen for him by his own people.”
—Excerpt from LotR chapter The Houses of Healing
“‘I have looked in the Stone of Orthanc, my friends...’
...He drew a deep breath. ‘It was a bitter struggle, and the weariness is slow to pass. I spoke no word to him, and in the end I wrenched the Stone to my own will. That alone he will find hard to endure. And he beheld me. Yes, Master Gimli, he saw me, but in other guise than you see me here. If that will aid him, then I have done ill. But I do not think so. To know that I lived and walked the earth was a blow to his heart, I deem; for he knew it not till now. The eyes in Orthanc did not see through the armour of Théoden; but Sauron has not forgotten Isildur and the sword of Elendil. Now in the very hour of his great designs the heir of Isildur and the Sword are revealed; for I showed the blade re-forged to him. He is not so mighty yet that he is above fear; nay, doubt ever gnaws him.’
‘But he wields great dominion, nonetheless,’ said Gimli; ‘and now he will strike more swiftly.’
‘The hasty stroke goes oft astray,’ said Aragorn. ‘We must press our Enemy, and no longer wait upon him for the move.”
—Excerpt from LotR chapter The Last Debate
‘Yet maybe this will lighten your heart,’ said Galadriel; ‘for it was left in my care to be given to you, should you pass through this land.’ Then she lifted from her lap a great stone of a clear green, set in a silver brooch that was wrought in the likeness of an eagle with outstretched wings; and as she held it up the gem flashed like the sun shining through leaves of spring. ‘This stone I gave to Celebrían my daughter, and she to hers; and now it comes to you as a token of hope. In this hour take the name that was foretold for you, Elessar, the Elfstone of the House of Elendil!’
—Excerpt from LotR chapter Farewell to Lórien