“Thee” and “thou” in “The Lord of the Rings”

[Inziladun]

Originally Posted by Formendacil

Denethor has also been brought up (his use of "thee" & co. to Gandalf is a clear sign that he considers Gandalf his inferior), and (again, IMS) I believe that Tolkien also mentions, in this or some other letter, that, among other things, he was using the archaic familiar form with the Gondorians to help establish a linguistic difference between Gondor and the Shire. He mentions that a big part of the reason people called Pippin "ernil i pherrianath (sic?) is because Pippin's Eriadoric Westron used only the familiar. Via the translator conceit into modern English, this doesn't play quite as well, since the surviving pronoun is the formal "you," but we can still get a hint of different "sound" of the two Westrons.

That's actually from the ROTK Appendix F.

Tolkien notes there, that in comparison to Hobbits:

In Gondor and Rohan a more antique language was used, more formal and more terse.

So obviously Éowyn would have been familiar with the formal pronouns, probably more so since she was raised by Théoden, whose mother was from Lossarnach.