According to Tolkien's letters, one of the Catholic elements he added to LotR unconsciously was any resemblance between Galadriel and Mary. In fact, it had to be pointed out to him by a reader after the book was published. He emphasizes that they're not the same, as he does with anything that might threaten to turn the story into an allegory in some minds. But he does agree that much of the imagery he uses for Galadriel probably came from that used for Mary. I find the imagery even more striking in the Elven song to Elbereth in "Three Is Company." If you take out the part about sowing the stars, and change the West to heaven, it could almost be a Marian hymn. And the Valar - including Elbereth - because they chose to enter physical creation, do have a physical aspect to them, unlike angels. But Elbereth isn't a parallel to Mary, either. About the only thing the three Ladies have in common is that none of them is divine.
I don't think (and neither did at least one of my professors) that it's a coincidence that the Christian traditions that pay the most attention to Mary are those that also have the most affinity for incarnational and sacramental spirituality - that is, for seeing the spiritual through the physical - because Mary is the incarnate link to the Incarnation. Matthew and Luke, the two evangelists that give us infancy narratives, talk about the virgin birth to emphasize not only that this birth was specially ordained by God, but that the child born is, in fact, divine. The mystery, of course, is that Jesus isn't half human and half divine, but completely both.
So it's not too surprising that Mary's most important role in Christian theology has been to hold together the two sides of the paradox of the Incarnation. It's also not surprising that it took the early Church a long time to figure out exactly what to think about the Incarnation and to come to terms with it being a paradox. Like hobbits, we humans like to be able to lay things out in an orderly fashion; we'll generally exhaust all other options before admitting that we can't completely understand something (remember Thomas Aquinas walking on the beach?). The following paragraph puts decades - or even centuries - of theological debate into a few sentences, so obviously isn't telling the whole story; but the whole story is fascinating to read about, if you like that kind of thing.
Some in the early Church tried to solve the problem by saying that Jesus was God and only seemed to be human. A major problem with this is that it makes Christ's death not a real death and, therefore, the Resurrection not a real resurrection; it would, in essence, not be any different from the "dying god" stories in many early mythologies. The opposite approach was to say that Jesus was simply human, but was "adopted" by God in a unique way at his baptism by John (that event chosen because of God proclaiming "This is my beloved Son"). But how could someone who was only human in nature heal the rift between God and Men? Somewhere in the middle (or off in left field) was the idea that Jesus was really two persons: one human and one divine. But that would create more problems than it would solve.
So the early Christian community finally threw up its hands (metaphorically speaking) and said, "We have to admit we don't understand it, but Jesus is one person with two natures - human and divine." And they found a way to define this both theologically and humanly by giving Mary the title "theotokos" or "God-bearer" or "mother of God." She's the mother of a person: Jesus of Nazareth. That person has two natures: human and divine. And "mother of God" has been Mary's most important title ever since - centering, as all her theologically-correct titles do, not on her but on her Son.
She's been given a lot of other titles since then; there's an entire litany of them (literally), and I'll admit that some of them make me say, "Huh?" I think people kept adding new ones that made sense to them, without deleting ones that no longer "worked" culturally. But they come from a tradition rich in imagery, and for someone with an active sense for images, such as the young Ronald Tolkien, that litany would have planted a lot of seeds that could have easily come to light - unconsciously, as he admits - in his imagery for Galadriel and, I would add, for Elbereth.
My own favorite imagery for Mary wasn't used by Tolkien for either of his Ladies, but it's very pertinent to LotR. It compares Mary to the dawn and Christ to the sun. Although, chronologically, dawn arrives before the sun, the sun is the source of its light. This is human-friendly imagery of something that takes place within the Eternal Now; as with all holy people who preceded Christ historically (including our fictional holy pre-Christians in LotR), the source of Mary's holiness is the Son who comes after her in time.
Speaking of theological traditions that have an affinity for physical statements of faith, if you think Western Rite Catholics pay a lot of attention to Mary, check out the Eastern Church! One of the gifts Eastern Christianity has given to all of us is the creation of icons. More than just a piece of art, an icon is a visible prayer. In traditional terminology, an icon isn't "painted" but "written," and the artist goes through a period of prayerful preparation for the act as one would for any deeply spiritual endeavor. There's a specific language to the meanings of symbols used in icons so that they can, indeed, be "read." Less objectively, there's something indefinable about icons, probably flowing from the prayerful manner in which they are created, that draws one into meditation and prayer (if one allows them to). The word "iconoclast" was originally used for people who considered the use of icons as aids in prayer to be idol worship, so went around destroying them.
The icon to the left is a version of one I've seen throughout my life, first on my grandmother's wall, then my mother's, and now my own, called "Mother of Perpetual Help." It shows a very human, but also divine, story. The young Jesus has a vision of angels bearing the instruments of His death (a Byzantine cross, as well as nails, a crown of thorns, a spear, and a branch with a sponge on it). The vision frightens Him, so He runs to His mother. You can see how fast He was running, because His sandal is dangling from His foot (I've seen a lot of different versions of this icon, and every one of them has had that dangling sandal). Mary holds Him, but she doesn't grasp Him to herself; as in most icons of Mary and Jesus, she's in a pose that seems to be offering Him to us. His hand is resting on hers, but she doesn't close her fingers over His; He doesn't belong to her. She's there to comfort and love Him, but she can't protect Him - or herself - from the future.
I'm not well enough versed in reading icons to know what all the details mean. The colors of clothing have significance, as do the sets of Greek letters. They're a little hard to see, but the letters near Jesus are almost certainly the Greek letters standing for "Jesus Christ, Son of God, Savior."* Jesus' halo is quite different from Mary's. Hers is a simple circle and seems to be transparent. Jesus' is not only opaque (you might even say solid), but also has rays that form a cross, somewhat difficult to see in the small picture. Mary is holy, but only Jesus is divine. In almost any classical painting - icon or otherwise - that shows Jesus with any saints and/or angels, His halo is significantly different from theirs, with that imagery making an essential theological statement. During most of the history of the Church, when few people could read, the halos in this icon would have been as clear a statement of Mary's non-divinity as the words written in a book would be for us. (For a different statement, look at the icon on the main contents page of these essays. This icon is a representation of the Holy Trinity, and all three Persons have equally opaque halos.)
There's a similar indication in the "Litany of Mary" mentioned earlier. The Litany of Mary (as well as the Litany of Saints, which starts with Mary and then goes through quite a list of them) begins with calling on God, in each of the three Persons of the Trinity. The response to each of those invocations is "Have mercy on us." But when the litany shifts to "Holy Mary, Mother of God," the response becomes "Pray for us." Mary can pray for us, as can anyone in heaven, on earth, or in purgatory, but granting mercy is a prerogative of God. I can't deny that there are some individual Catholics who, for whatever reason, seem to consider Mary divine, but Church teaching has never made her so.
In the same way, Tolkien says very clearly that the Valar are not divine. Galadriel certainly wasn't, being an elf, especially one who had been rebellious. Tolkien says, in fact, that the Valar are purposely meant to be acceptable "for one who believes in the Blessed Trinity." Since Tolkien's subcreation is pre-Christian, we see there only half of the paradox of the Trinity: The One. But that doesn't mean that The Three aren't an unseen part of it.
*These letters that form the acronym for "Jesus Christ, Son of God, Savior" in Greek are the source of the Icthus - or fish symbol - for Christianity (Icthus being not only the acronym but also the Greek word for fish). The symbol is most often seen in the version formed by two simple intersecting curved lines, which is used by a variety of Christians on stationery, jewelry, etc. The story behind this particular form of the symbol is that during the persecution of Christians under the Roman Empire, when someone wanted to be sure the person they were talking with was a fellow Christian, they would draw the first arch of the fish on the ground - something that would be seen by a non-Christian as a casually-made mark in the dirt/sand. If the second person was Christian, he or she would complete the symbol by drawing the second arch, and the first person would know it was safe to talk about their faith. I can't give a specific historical reference for this, but I've heard it often enough that I suspect it's true. It's important to recognize that this would have been a matter of safety rather than of a shared "secret knowledge" as it might have been among gnostics. Later in history, artists would sometimes use a more realistic-looking fish as a symbol for Christ. One place I've noticed it is on small stained-glass windows, where the artist wanted to depict Christ but didn't have room for a human figure.