LotR -- Book 5 - Chapter 02 - The Passing of the Grey Company

[Celuien]

'Too often have I heard of duty,' she cried. 'But am I not of the Houseof Eorl, a shieldmaiden and not a dry-nurse? I have waited on faltering feet long enough. Since they falter no longer, it seems, may I not now spend my life as I will?'
'Few may do that with honour,' he answered.

Theoden seems to have strongly impressed duty onto Eowyn. With the high value placed on duty and honor, he would have had no reason to think that Eowyn would abandon her charge, even if te thought had crossed her mind. I agree that her belief that Aragorn was going to his death helped spur her on to the decision; perhaps it was also related to the realization that Aragron did not return her love.

At any rate, it was fortunate that she did finally choose to leave Edoras.