LotR -- Book 5 - Chapter 06 - The Battle of Pelennor Fields

[Celuien]

Éowyn's courage is a highlight in this chapter. I think there are several factors that bring out her fearlessness.

Hopelessness is a major factor. For the last few chapters, she has been looking for death, and it is that impression which remained with Merry during their ride together. But even more than despair, I think Éowyn draws on love for Théoden.

'Begone, foul dwimmerlaik, lord of carrion! Leave the dead in peace!'
A cold voice answered: 'Come not between the Nazgûl and his prey! Or he will not slay thee in thy turn. 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 to the Lidless Eye.'
A sword rang as it was drawn. 'Do what you will; but I will hinder it, if I may.'
'Hinder me? Thou fool. No living man may hinder me!'
Then Merry heard of all sounds in that hour the strangest. It seemed that Dernhelm laughed, and the clear voice was like the ring of steel. 'But no living man am I! You look upon a woman. Éowyn I am, Éomund's daughter. You stand between me and my lord and kin. Begone, if you be not deathless! For living or dark undead, I will smite you, if you touch him.'

Her answer to the Witch King isn't resignation to death or defeat, but one of defiance in defense of her "lord and kin." So while her journey to battle may have been born out of loss of hope, here at least I think she is acting not only out of the recklessness of despair, but also out of both her upbringing in Rohan and a love that outweighs any fear. All three acting together are an amazing combination.