Passages from Tolkien's first drafts of LotR, from The History of Middle Earth series edited by Christopher Tolkien. Parts in italics are Christopher Tolkien's commentary, the rest is Tolkien's writing. I spent a long time typing these up; please don't steal.

From the first draft of "The King of the Golden Hall"
Eowyn Elfsheen daughter of Eomund?

Very fair and slender she seemed. Her face was filled with gentle pity, and her eyes shone with unshed tears. So Aragorn saw her for the first time in the light of day, and after she was gone he stood still, looking at the dark doors and taking little heed of other things.

Aragorn was silent, but his eyes followed Eowyn

Long she looked upon Aragorn, and long he looked upon her

And after Theoden’s words, Aragorn says: ‘If I live, I will come, Lady Eowyn, and then maybe we will ride together.’ Then Eowyn ‘smiled and bent her head gravely.’

Aragorn weds Eowyn sister of Eomer (who becomes Lord of Rohan) and becomes King of Gondor.

Tolkien's notes on the chapter:
? Cut out the love story of Aragorn and Eowyn. Aragorn is too old and lordly and grim. Make Eowyn a stern amazon woman.

If so, alter the message of Galadriel:
Elfstone, Elfstone, bearer of my green stone,
In the south under snow a green stone thou shalt see.
Look well, Elfstone! In the shadow of the dark throne
Then the hour is at hand that long hath awaited thee.

The green stone was borne on Theoden’s brow (a diamond in the final version), and Eowyn would stand "in the shadow of the dark throne."

Probably Eowyn should die to avenge or save Theoden.

But my father added in a hasty scribble the possibility that Aragorn did indeed love Eowyn, and never wedded after her death.

Random passages and notes:
They strike the Dunharrow ravine on the second day. Aragorn rides with Eowyn.

Aragorn had already arrived and greets King Theoden side by side with Eowyn.

Aragorn and Eowyn meet the King. They say that Riders are mustering at Dun Harrow.

Eowyn says that women must ride now, as they did in a like evil time.

On either side of the King were Eowyn and Eomer, and Aragorn beside Eowyn.

Tolkien's different ideas on the Battle of the Pelennor; he originally planned for Eowyn to die:
(I) Feb 15 – Eowyn slays the King of the Nazgul and is mortally wounded.

(II) Feb 3 – Death of Theoden and Eowyn in killing the Nazgul King.
Feb 7 – Return to Gondor. Crowning of Aragorn. Funeral of Theoden and Eowyn.

(III) Feb 11 – Theoden and Eowyn destroy Nazgul and Theoden falls.

Theoden falls from horse sorely wounded; he is saved by Merry and Eowyn, but sortie from Gate does not reach them in time, before Eowyn is slain.

Original version of Eowyn slaying the Witch-King:
But Theoden was not alone. One had followed him: Eowyn daughter of Eomund, and all had feared the light of her face, shunning her as night fowl turn from the day. Now she leapt from her horse and stood before the shadow; her sword was in her hand.

"Come not between the Nazgul and his prey," said a cold voice, "or he will bear thee away to the houses of lamentation, beyond all darkness where thy flesh shall be devoured and thy shrivelled mind be left naked."

She stood still and did not blanch. "I do not fear thee, Shadow," she said. "Nor him that devoured thee. Go back to him and report that his shadows and dwimor-lakes are powerless even to frighten women." The great bird flapped its wings and leapt into the air, leaving the king’s body, and falling upon her with beak and claw. Like a shaft of searing light a pale sword cold as ice was raised above her head.

She raised her shield, and with a swift and sudden stroke, smote off the bird’s head. It fell, its vast wings outspread crumpled and helpless on the earth. About Eowyn the light of day fell bright and clear. With a clamour of dismay the hosts of Harad turned and fled . . . Eowyn stepped to the king. "Alas, Theoden son of Thengel," she said. "But you have turned the tide. See, they fly. The enemy is broken by fear. Never did an old Lord of Men die better. You shall sleep well, and no Shadow nor foul thing assail your bed."

There is no suggestion that Eowyn was in any way hurt in the encounter with the Lord of the Nazgul or after . . . no part was foreseen for Merry in the great event.

First version of "The Last Debate":
Gimli, Legolas, and Pippin talk. They hear of the love of Eowyn for Aragorn in Dunharrow.

"So they told us," said Legolas, "and they forbade us to go in; but Aragorn could not be turned from it. He was in a grim mood. And that fair lady that lies now in the Houses below, Eowyn, wept at his going. Indeed at the last in the sight of all she set her arms about him, imploring him not to take that road, and when he stood there unmoved, stern as stone, she humbled herself to kneel in the dust. It was a grievous sight."

"But do not think that he was not moved," said Gimli. "Indeed, I think Aragorn himself was so deeply grieved that he went through all perils after like a man that can feel little more. He raised her up and kissed her hand, and then without a word we set out."