How is Eowyn, shield maiden of Rohan written in Rohirric (is that the language of Rohan?)?
Edit - this answer is wrong
I've located a note in HoME12 stating that the East Gates of Moria "had borne Runic inscriptions in several tongues ... commands that all should depart who had not the leave of the Lord of Moria written in Quenya, Sindarin, the Common Speech, the languages of Rohan and of Dale and Dunland".
This makes it clear that (1) Rohirric was written, and (2) these inscriptions were obviously meant to be read by passers-by, so the Rohirrim in general could read.
Therefore I need to withdraw this answer.
I'm leaving it in it's original state below nonetheless as it may be useful to indicate that it's an incorrect interpretation of Aragorn's words in TT. But no more upvotes, please, and I'll gladly accept any downvotes anyone feels it necessary to give.
It's most likely not.
Rohan is very probably a pre-literate society, per Aragorn's remarks in the Two Towers:
They are proud and wilful, but they are true-hearted, generous in thought and deed; bold but not cruel; wise but unlearned, writing no books but singing many songs...
There are only two references to any kind of writing in connection with the Rohirrim that I can find in LotR; the first one describing the floor of the Golden Hall:
As their eyes changed, the travellers perceived that the floor was paved with stones of many hues; branching runes and strange devices intertwined beneath their feet.
And the second describing the horn that was given to Merry:
Then Éowyn gave to Merry an ancient horn, small but cunningly wrought all of fair silver with a baldric of green; and wrights had engraven upon it swift horsemen riding in a line that wound about it from the tip to the mouth; and there were set runes of great virtue.
In the first case, it need not be so that the "runes" were actual writing; it could be just abstract symbols (which would be consistent with "...and strange devices", and similar to that described for the Druedain in Unfinished Tales: "They came no nearer to writing by their own invention than the use of a number of signs, for the most part simple, for the marking of trails or the giving of information and warning") or it could be a different language.
In the second case the horn was explicitly described as having been made by Dwarves in the following paragraph.
Rohirric could probably be adapted to a writing system (such as the Feanorian letters), and no doubt some of the more learned of the Rohirrim (such as Theoden, who spent time in Gondor during his youth) can read and write other languages, but Rohan would have no writing system of it's own.
We can't be absolutely sure. There are two principal issues here:
- What is Éowyn's Rhorric name?
- What writing system did the Rohirrim use?
An important thing to remember about Tolkien is that, in-universe, he's translating from other languages into English; so the name "Éowyn" is not the name she would have been known by in-universe.
Unfortunately, although Tolkien wrote a great deal about his fictional languages, he didn't give us enough to translate her name back into Rohirric. The best we can do is translate the "Éo-" prefix, which Tolkien describes as:
The element éo-, which so often appears (not unnaturally, being an old word meaning 'horse', among a people devoted to horses), represents an element loho-, lô- of the same sense.
History of Middle-earth XII The Peoples of Middle-earth Chapter 2: "The Appendix of Languages"
According to Appendix E, the Rohirrim wrote in a variation of the Cirth, similar to the runes used by the Dwarves:
The Cirth in their older and simpler form spread eastward in the Second Age, and became known to many peoples, to Men and Dwarves, and even to Orcs, all of whom altered them to suit their purposes and according to their skill or lack of it. One such simple form was still used by the Men of Dale, and a similar one by the Rohirrim.
Return of the King Appendix E "Writing and Spelling" II: "Writing"
Unfortunately, we don't know what this simplified Cirth would look like; the only Cirth alphabet provided by Tolkien is the one in Appendix E, the full Angerthas Daeron of the Noldor.
What can we do?
Not much. If you wanted to write the word "Éowyn" in the Cirth we're given, you can do that; it would look something like1:
We could try the loho-/lô- prefix of her Rohirric name as well, but without knowing how the rest of the name would modify it, it wouldn't be a terribly profitable exercise.
1 Image courtesy of theHobbit-movie.com's rune generator