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:
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:
And the second describing the horn that was given to Merry:
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:
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:
According to Appendix E, the Rohirrim wrote in a variation of the Cirth, similar to the runes used by the Dwarves:
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