The name "Éowyn" is made up of two letters from Anglo-Saxon Futharc (a runic alphabet).
It's well known in LotR circles that Tolkien utilized a lot of existing languages, histories and mythologies in creating Middle-Earth. You can check out some of these inspirations in the writings section.
Tolkien himself was a linguist. His languages of the elves (Quenya and Sindarin), for example, were modeled after the Finnish and Welsh languages.
The language of Men, and especially the language of Rohan, was modeled after English. In fact, many proper names in Rohan have parts directly from Old English. In the writings section, there's an article that describes some of these names.
About the Runes
The runic alphabets developed in the Germanic or northern regions of Europe, and were used from the first century all the way through the middle ages.
Their source is still a big huge mystery, but most linguists agree that the designers of the runes were familiar with both the Latin and Etruscan languages (or at least their alphabets).
The earliest form of the runes is called "Elder Futharc." The name "Futharc" comes from the first six letters of the runic alphabet, much as our own name "alphabet" comes from the first two letters of the old Greek alphabet--alpha and beta.
Different versions of this "Elder Futharc" evolved throughout Europe. Two in particular were Anglo-Saxon Futharc and Younger Futharc (used widely in medieval Scandinavia). But there are also Hungarian and Turkish forms of the runes, all based on the original Elder Futharc.
Tolkien used Anglo-Saxon Futharc (combined with some Younger Futharc) to create his own runic alphabet, called "Cirth." From this alphabet, he was able to offer written samples of Dwarvish, Quenya and Sindarin within the text of "The Lord of the Rings."
The Puzzle of Rohan
Interestingly enough, Tolkien did not use his own Cirth alphabet in naming the people and places of Rohan. Instead, he used patterns found in the Anglo-Saxon form of Futharc and later patterns found in Old English to create "Rohirric".
Rohirric, the language of the Rohirrim, is based on Old English and Anglo-Saxon Futharc. In Middle-Earth, though, Rohirric is closely related to "Westron," the common tongue (equal, of course, to Modern English), as well as to the Hobbit languages.
Notes on Etymology:
1. Éowyn has no parts derived from the dative 'to you' form [eow].
2. eoh and wyn are from Anglo-Saxon Futharc, the ancestor of Old English. In other runic alphabets, the letters would be different. In Elder Futharc for example, the eighth and nineteenth letters, while still retaining the same meaning, are ehwaz and wunjo.
3. The rune [wyn] is thought to have numerous meanings, including: joy, comfort, laughter, and glory.
4. The pronunciation of "Éowyn" is a little tricky (if you want to be fanatical about it and do it correctly). Most people, including me, just pronounce it é-o-win. To be more accurate, the first syllable "éo-" should sound very close to "ear."
For more information:
Runes: Alphabet of Mystery - everything you ever wanted to know about the runes, and then some.
Omniglot: the Cirth - Tolkien's alphabet and its usage, as well as links to other "alternative" languages.
Alphabets of Middle-Earth - Dan Smith's site.
Council of Elrond: Languages - just follow the links; a great resource if you wanna learn more about Tolkien's languages.
Check out these books:
The English Language: A Historical Introduction by Charles Barber
Secrets of the Runes by Nigel Pennick
Nordic Runes: Understanding, Casting, and Interpreting the Ancient Viking Oracle by Paul Rhys Mountfort
Nine Doors of Midgard: A Complete Curriculum of Rune Magic by Edred Thorsson
Old English Grammar by Randolph Quirk & C.L. Wren
Word-Hoard: An Introduction to Old English Vocabulary by Stephen A. Barney
* Some information from "The English Language: A Historical Introduction" by Charles Barber and "Secrets of the Runes" by Nigel Pennick.