I truly and deeply love Eowyn because she's angry, bitter and can be genuinely cold when we meet her - and she's only 24! already she's changed by this war, by the loss of her parents, by the long period of her life where she had no parental figure because Theoden was battling his own depression, and by Grima being a creep. Tolkien's young characters go to war and are changed by it, like Pippin, like Sam, who is still quite a young hobbit (he's 39, and they come of age at 33), but not her - she is already deep in grim thoughts about dying for honour, cheerless, cold.
I don't like comparing all Tolkien women to one another because they are quite different from each other and comparing them just because they're women often feels reductive because they serve vastly different roles in the story, but when you're considering how he presents femininity, it's necessary to do so. so far we've met Lobelia, Mrs. Maggot, Goldberry, Arwen, Galadriel and Eowyn (Rosie was only mentioned as far as i remember?). hobbit women we meet while we are still at home, and they fit right into the homey atmosphere of the shire, in which characters are often presented playfully, or have one defining trait (think about the characters we only know from notes attached to bilbo's "gifts"). further from home, we meet goldberry, arwen and galadriel, who are old, fair, good, beautiful. each of them is different (and galadriel especially has a dark, flawed part of her we can see), but with the women meet on the journey, further from home, a pattern starts to emerge - they come from a different time, from a different world, and even with all their fairness and kindness they can at times feel distant, out of reach.
and then!! eowyn comes into the picture, and- she could be you. she could be me. she's not only complex, but also within reach. she's not a gentle or joyful presence, she's not a powerful ancient force, she doesn't come bearing gifts for everyone - she's so full of negative emotions and pain which she needs to heal from, she's so young and already feels like an old, weary soul.
and it's not to say one kind of character is better than the other, or more complex or anything, because that's exactly the reductive way of looking at those women that I don't like seeing in discussions. ultimately, they are all just different people. but the introduction of eowyn broadens the spectrum of femininity shown in LOTR, and while it's still not exactly wide, it's a bit wider than it was before.
