Aragorn tells Eowyn that she can’t come with him on The Paths of the Dead because her people need her and that renown isn’t really all it’s cracked up to be. He’s not wrong, exactly, but he basically tells her it’s her duty to stay behind, something he would never say to her uncle or brother. And she calls him on it. Flat out. She tells him, “All your words are but to say: you are a woman, and your part is in the house. But when the men have died in battle and honour, you have leave to be burned in the house, for the men will need it no more. But I am of the House of Eorl and not a serving-woman. I can ride and wield blade, and I do not fear either pain or death.” Think about that for a moment. Not only is she calling him out for sexism, she lays out why it’s sexist and does a pretty damn fine job of distilling down the lot of women in this culture. To whit: if there aren’t men around, you don’t really matter, and you definitely don’t get to decide for yourself how you live OR die if you’re a lady. That’s very powerful, especially in a series that deals a lot with the trappings of war and glory from a distinctly masculine point of view.
Except to take all of the blood line of a ruling family into battle is stupid. To lose the entire power structure at once would be catastrophic. It was incredibly selfish and immature of her to do so.
I think we can waive that as the guy with the irreplaceable blood of Numenor and last hope of humanity is the one telling her no. If he dies, Gondor is screwed. In setting, Theodin, Eomer, and Eowyn are all replaceable. Aragorn himself? Not so much. This is the pot calling the kettle black. He’s already doing something incredibly stupid, taking her with him isn’t going to make it any worse or make it any worse for Rohan.
If we’re also going by real world rules then while Eomer fighting in the battle is understandable, it’s just as selfish, immature, and stupid.
“Behold! I go forth, and it seems like to be my last riding,” said Theoden. “I have no child. Theodred my son is slain. I name Eomer my sister-son to be my heir. If neither of us return, then choose a new lord as you will. But to some one I must now entrust my people that I leave behind, to rule them in my place. Which of you will stay?”
No man spoke.
“Is there none whom you would name? In whom do my people trust?”
“In the House of Eorl,” answered Hama.
“But Eomer I cannot spare, nor would he stay,” said the king; “and he is the last of that House.”
“I said not Eomer,” answered Hama. “And he is not the last. There is Eowyn, the daughter of Eomund, his sister. She is fearless and high-hearted. All love her. Let her be as lord to the Eorlingas, while we are gone.”
“It shall be so,” said Theoden. “Let the heralds announce to the folk that the Lady Eowen shall lead them.”
1) Theoden does not even remember she exists as a member of his House and viable candidate for leadership until Hama reminds him, even though that’s what she was doing when Theoden was under Wormtongue’s influence.
2) If someone is being stupid, it’s a family trait. Her uncle and her brother are guilty of the same flaw. The only reason she’s not allowed to go is because she is a girl, which Tolkien points out in the quote above.