It’s no great leap to look for imaginative links between Albus Dumbledore, one-time headmaster of Hogwarts School of Witchcraft and Wizardry, and Gandalf, the Grey Pilgrim of Middle-earth. Both are major characters in their respective fictive worlds, both are immensely powerful but equally humble wizards, both have an unexpected sense of humor, both are the primary tacticians against an evil Dark Lord, both sacrifice their lives for the greater good (Gandalf returns — for a time; will Dumbledore?) — and so forth. But what I’d like to explore today relates to an interesting philological nexus between the two of them.
Tolkien once said, “I always in writing start with a name. Give me a name and it produces a story, not the other way about normally” . The emergence of Gandalf into Tolkien’s imagination is certainly one of the best examples of this “creation from philology” (to use Tom Shippey’s term). This subject been explored elsewhere and at greater length, so I’ll just sum up for those of you unfamiliar with the background. The name Gandalf (modernized from Gandálfr) appears in several places in Old Norse literature, but most particularly in the Dvergatal (the “Catalog of Dwarves”, if you will) — a section of the Eddic poem, the Völuspá (“The Song of the Seeress”). The word comes from two distinct Old Norse elements, gandr “wand, staff” and álfr “elf” — and the suitability of such a name to the character we now know as Gandalf is pretty clear.  As to the specific way in which Gandalf is described in The Hobbit, there was another source of inspiration for Tolkien. The story (as related by Carpenter in his biography of Tolkien) goes like this: on a walking tour of the Swiss Alps in 1911, Tolkien purchased a postcard reproduction of a Josef Madlener painting, Der Berggeist (“the Mountain Spirit”), which depicted a rather Gandalf-like figure (with arguably some Radagast thrown in) — as you can see at the top of this post. Douglas Anderson explains some of the problems with this story, but however it really transpired, the fact remains that at some point, Tolkien came by this postcard and noted on it, years later, “Origin of Gandalf.” 
So, Gandalf, if not actually an elf himself, is associated with them through a name that essentially means “magical elf.” What about Albus Dumbledore?
Let’s dispense with the cognomen first. Rowling herself has pointed out that dumbledore is a British dialectal word for a “bumblebee”, and that she chose it because she imagined Dumbledore buzzing about Hogwarts Castle like a bee. If one takes the time to look, one finds this word in several, mainly southern British dialects — for example, the Kentish , Sussex , and Gloustershire  dialects, inter alia. Its etymology appears to be onomatopoeic (so, too, the “bumble” in bumblebee). Tolkien himself uses the word (spelled slightly differently) in this sense in the poem “Errantry”:
He battled with the Dumbledors,
the Hummerhorns, and Honeybees,
and won the Golden Honeycomb [...]
As a side note, Tom Shippey has suggested there may be a connection between Dumbledore and C.S. Lewis’s Professor Dimble in That Hideous Strength . There certainly may be; the similarities appear to be more than superficial.
Moving on, what of Albus? The name is Latin, coming directly from albus “white” (with the Greek cognate αλφος). There is, for instance, the Liber Albus, the White Book of the City of London, compiled in 1419 by John Carpenter and describing the laws and customs of 15th Century London. Against the White Book, I suppose we might set Tolkien’s Red Book of Westmarch, not to mention the Golden Book of Tavrobel. But before I get too far afield ...
The Latin albus gives rise to English albino, as well as alb, a long white religious vestment. And the words alp (as in the Swiss Alps) and alpine derive from the same source, too, in reference to their white, snow-capped peaks  — which brings us back to Tolkien’s walking tour of the Alps. Well, Albus Dumbledore certainly is a white figure (like Gandalf, who explicitly becomes Gandalf the White) — both in terms of his age and appearance, and in terms of his diametric opposition to the Dark Lord, Voldemort, the representative figure of blackness and evil. Also, Albus Dumbledore is a distinctly British figure, of course; well, we should remember that one of the most ancient names of Britain was Albion, in reference to the White Cliffs of Dover (though actually, this name was apparently adapted from a Celtic source with a related, but slightly different meaning). I think that’s a nice touch in a fictive world so strikingly English. Speaking of Albion, there is also the historical figure of Alboin, whose name is the Lombardic form of Ælfwine (“elf-friend”), and whom Tolkien adapted for use in his legendarium . I won’t dig any deeper into that here, but it offers another direction to explore at some time in the future.
And another side-note on Albus qua “the White”: Hagrid’s first name, you’ll remember, is Rubeus. This is from Latin rubeus “red, reddish” — certainly suitable for a half-giant who enjoys his drink. The 18th Century Italian historian, Ludovico Antonio Muratori, actually puts these two in close proximity in one his dissertazioni, where he wrote, “pulcherrima divisa est color albus, et rubeus” . Does this lead us anywhere? I’m not sure — maybe; but it’s interesting ...
And there’s more. Remember that álfr (in Gandalf) means “elf” — well, Latin albus is its source! “Elf” seems to have originally meant something like “the white one” , a supernatural being with a bright, shimmering, illuminated aspect. Both Old Norse álfr and Old English ælf (not to mention the Old Mercian form, elf, as well as Old High German alp, alb, Old Frisian alf, and even Old Irish ailbhín “flock”) derive from a hypothetical Germanic root *albiz, which came directly from Latin albus, Greek αλφος (which in turn trace their source to Indo-European *albho “white”).
So what have we learned here? That Albus Dumbledore and Gandalf may very well have a common etymological connection: the association with magic, elves, and nature. We know this was all by design in Tolkien’s works. How much of this was intentional on Rowling’s part? Well, certainly Latin albus “white” would have been. My guess is that albus > ælf, elf was probaly not, though one can still attempt to make the case — remember how friendly and sympathetic Dumbledore is toward Dobby? :)
Are any of you still with me? This has been one of my longer posts — I thought about breaking it up into parts, but I wasn’t quite sure where to place the fault line. I’ll very likely develop this into a more formal treatment (a conference paper, most likely) at some point. Any thoughts?
 Radio interview with J.R.R. Tolkien, conducted by Dennis Gerrolt for the BBC radio program “Now Read On ...”, January 1971.
 Shippey, Tom. Roots and Branches: Selected Papers on Tolkien. Berne: Walking Tree Publishers, 2007, pp. 195-6.
 Tolkien, J.R.R. The Annotated Hobbit. Ed. Douglas Anderson. Boston: Houghton Mifflin, 2002, pp. 36-9.
 Paris, W.D. A Dictionary of the Kentish Dialect and Provincialism in the County of Kent. Lewes: Farncombe & Co., 1888, p. 48.
 ———. A Dictionary of the Sussex Dialect and Provincialism in the County of Sussex. Lewes: Farncombe & Co., 1875, p. 38.
 Robertson, J. Drummond. A Glossary of Dialect and Archaic Words used in the County of Gloucester. London: English Dialect Society, 1890, p. 41.
 In an unpublished essay and in private correspondence.
 Taylor, Isaac. Names and Their Histories: A Handbook of Geography and Topographical Nomenclature. London: Rivingtons, 1898, p. 43.
 Shippey, pp. 269-71.
 Dizionario di Erudizione Storico-Ecclesiastica, Vol. XXIII. Venezia: Emiliana, 1843, p. 154.
 Hall, Alaric Timothy Peter. The Meanings of Elf and Elves in Medieval England. Ph.D. dissertation, 2004, pp. 56-7.