It's something I've been debating too! Looking at modern monarchy, marrying a prince DOES make you a princess, but I'm trying to figure out if a "born royal prince" would be considered different to a "made a prince" in terms of hierarchy and inheritance titles. (co-thought: was Elboron a Prince?)
Personally I am a "rule of cool" type person. So to me it would be cool if Éowyn either became a princess at her brother's crowing, or at her marrying Faramir. BUT I would also very interested to hear any more thoughts you have on the topic!
Ok obligatory statement: I'm so sorry its taken me like two months to answer this! I loved this question in my inbox when you sent it, and I've thought a lot about it since, but the holidays/moving chaos made it difficult for me to sit down and find a time to finish my draft response! So please pardon the delay!
That said, here are my (slightly disorganized) thoughts:
So to go a bit out of order, I do think Elboron is definitely a Prince- he inherits the title of Prince of Ithilien in much the same way that the title of Prince of Dol Amroth is hereditary in Imrahil's line. In The Steward and the King, this isn't stated directly exactly, but I think it's pretty much a foregone conclusion from the information we do get in that chapter.
Indeed, in regard to the office of Steward, Aragorn says the office shall belong to Faramir and his heirs, for as long as Aragorn's line lasts:
And regarding the Ithilien, we learn:
So while Aragorn doesn't exactly say "the princedom belongs to you and your heirs", I think thats implied. And in History of Middle Earth, vol. XII (The Peoples of Middle Earth) I believe we do get confirmation that Elboron and his heirs continued to be both Stewards and Princes of Ithilien by way of JRRT's abandoned Fourth Age sequel. I don't have a PDF or the book in front of me, so I can't give you a quote at the moment, but I can circle back to that later.
But all of that is to say that yes, Elboron certainly inherits both (all?) of his father's titles.
As for the main question, "Is Éowyn a princess, and if so, when/how is that title applied to her?" I don't find any direct textual evidence that she held either of those titles. I totally agree with you about the Rule of Cool, I think "Éowyn, Princess of Rohan and/or Ithilien" is fuckin' rad and it's totally fair game to refer to her as such, even without direct textual evidence. Another point in that column is that LOTR is meant to be a found/translated history, and thus I think in addition to the Rule of Cool, we can always appeal to the "Rule of Ambiguity in Translation" and say "maybe Éowyn was indeed called Princess and that title didn't make it into the Red Book for some reason, who knows?!".
I think those are both totally fair and complete ways to answer this question. That said, being the academic that I am, I wanted to dig a little deeper and see if there was any indirect textual evidence that might point us to whether or not there's precedent for Éowyn to be called a princess by virtue of any of her positions and relations.
First off, lets think about her role in Ithilien. The book is quite clear that Éowyn's titlular counterpart to Faramir's "Prince of Ithilien" is "Lady of Ithilien". She could have been named Princess of Ithilien as Faramir's wife, but she isn't. So if she is called a princess, I think it's probably not because of being married to the Prince of Ithilien.
What about Rohan? I think one way for her to be a princess of Rohan is through her mother, Théodwyn, who was the daughter of a king and thus could have plausibly been called Princess herself. Another way is the way you mentioned, that she becomes a princess when her brother is crowned king. And I suppose a third way might be that Théoden at some point in his fostering of Éowyn and Éomer bestowed the titles of prince and princess upon them, respecitvely.
Unfortunately for the cool factor, I don't know of any historical precedent for any of these methods of title transference, at least not in in the cultures and times that inspired Tolkien's writings. Titles were generally transferred through the father's line in European history, so its unlikely that Éowyn would've been a princess from birth unless her father Éomund was called Prince. I see no evidence of this in the appendices or any of the texts. Éomund was certainly a nobleman- he was Lord of Aldburg- but nowhere is he called a prince. So it's highly unlikely such a title would have transferred to either of his children.
In terms of Théoden bestowing such titles on Éomer and Éowyn, I also don't really see why he'd do that while his son Théodred still lived and I know of no historical precedent for this. RE: Théodred, it's interesting because Théodred does seem to be called a prince a few times (Tolkien gateway refers to him as such) but in my copies of LOTR and Unfinished Tales, I don't see Théodred ever directly named as a prince (please correct me if I missed something, I want to know where in the text the title of Prince comes from for Théodred, if anywhere!).
Finally, would Éowyn be called Princess upon her brother's ascention to the throne of Rohan? No. Princess is not usually a title given to the sibling of a king. So I see no precedent for that.
Another point in the "no Princess title" column comes from the fact that Tolkien based the Rohirrim off of Anglo-Saxons.
If I recall my history correctly (please correct me if I do not!) female titles among Anglo-Saxon nobility were far less common than in later periods in Europe. For example, Aethelflaed, daughter of King Alfred the Great of Wessex is called "Lady of the Mercians", not "Princess", despite being the daughter of a king.
Even Alfred's wife, Eahlswith, was not called "Queen", but rather referred to as a "noble Mercian lady" and "chaste widow" (1). However, it is also worth noting that Tolkien does refer to Théoden's wife Elfhild as "his queen", and thus, there is precedent at least for the title of "Queen" in Rohan.
Ok, so I've given you a whole bunch of reasoning that basically says, Éowyn was probably not called Princess. Can I give you any reasoning to the contrary? Is there anything in the text that backs up our "Rule of Cool"? Actually yes. In my research for this post I found a few (albeit weak) lines of evidence.
One is that both Faramir and Boromir refer to Boromir as a "prince of the city" at different times, despite the fact that Boromir does not hold the legal title of Prince.
It is therefore plausible that even without holding the formal legal title of "Princess of Rohan and/or Ithilien", Éowyn might still be called a princess in a poetic, metaphorical, or informal sense.
Another weak line of evidence is that Ancalimë was referred to as a Princess in Númenor.
We know Gondor descended from Númenor, and while thousands of years stand between the two cultures, its possible that the idea of Princess as a title has stuck around and was simply not recorded in the texts that led to LOTR.
Finally, the Northmen progenitors of the Rohirrim called some of their noblewomen princesses, such as Vidumavi, a princess of the Northmen who married Valacar, the eventual twentieth king of Gondor. She is indeed named as such in Unfinished Tales.
So, similar to the Númenorian Princess argument, there's evidence that in proto-Rohirrim culture, the title of "Princess" was indeed applied to noblewomen, and likewise it may have also stuck around and simply not been recorded in the annals of LOTR.
So, what can we conclude? Well, unfortunately not much. I think that despite this very long post and a decent chunk of hyper focus research and reading for me, we're pretty much where we started: there's no substantial evidence that Éowyn is called Princess formally, from any direction. However, I don't think anything we've said truly rules out the possibility either. There are a few threads of cultural precedent that we can weave into a case for calling her princess if we want to do so.
So, in the end, I think there's no way to be perfect and pedantic here. Rule of Cool wins out.
Tl;dr: sure, Éowyn, Lady of Ithilien and Princess of the Riddermark, Wraithsbane and Wild Shieldmaiden of the North can and absolutely should be a thing.
