[Formendacil]

If memory serves, Tolkien comments on his use of Thee/Thou/Thy/Thine in one of his letters, but (also if memory serves, or IMS) he doesn't speak of Aragorn and Éowyn; rather, he speaks of Faramir and Éowyn. For those who know to look for it, Éowyn's "defrost" becomes evident in the 2nd person pronouns she uses in addressing Faramir.

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.

(As an aside, it is frustrating to try and say things with any certainty when one's copy of The Letters is a 1000 leagues distant.)