It is discussed at great length, whether or not the Arkenstone is one of the Silmarils1, and the most accepted answer is that it is not2.
The Arkenstone however, has undeniable 'powers' or properties which make it very special3.
So in light of this, what is the Arkenstone?
Has Tolkien ever written anything about it? Is there something I've missed from the books?
Edit:
There seems to be some new info stating that the Arkenstone is not a Silmaril.
The Arkenstone is definitely not a Silmaril for the following reasons (not an exhaustive list):
Ultimately, while it can be argued that Tolkien did derive inspiration from the Silmarils when inventing the Arkenstone (in particular see his use of Eorclanstanas to describe the Silmarils in the Old English versions of texts given in HoME4), inspiration is all that it was and the evidence is just too strong against it being one.
Everything that Tolkien ever wrote about the Arkenstone (aside from incidental mentions) is collected in the text of the Hobbit, so it's worth quoting all of the descriptive passages.
But fairest of all was the great white gem, which the dwarves had found beneath the roots of the Mountain, the Heart of the Mountain, the Arkenstone of Thrain. "The Arkenstone! The Arkenstone!" murmured Thorin in the dark, half dreaming with his chin upon his knees. "It was like a globe with a thousand facets; it shone like silver in the firelight, like water in the sun, like snow under the stars, like rain upon the Moon!"
This is just a descriptive passage and offers no clues to it's nature beyond it being a "great white gem".
It was the Arkenstone, the Heart of the Mountain. So Bilbo guessed from Thorin’s description; but indeed there could not be two such gems, even in so marvellous a hoard, even in all the world. Ever as he climbed, the same white gleam had shone before him and drawn his feet towards it. Slowly it grew to a little globe of pallid light. Now as he came near, it was tinged with a flickering sparkle of many colours at the surface, reflected and splintered from the wavering light of his torch. At last he looked down upon it, and he caught his breath. The great jewel shone before his feet of its own inner light, and yet, cut and fashioned by the dwarves, who had dug it from the heart of the mountain long ago, it took all light that fell upon it and changed it into ten thousand sparks of white radiance shot with glints of the rainbow.
This is the main passage that is probably responsible for much of the "Arkenstone = Silmaril" speculation, yet it fails on many of the points I list above.
The Elvenking himself, whose eyes were used to things of wonder and beauty, stood up in amazement. Even Bard gazed marvelling at it in silence. It was as if a globe had been filled with moonlight and hung before them in a net woven of the glint of frosty stars.
And this is a relatively minor passage that offers nothing much new.
In the end that's all we have to work with. There's nothing else in any of the books, nothing in his Letters.
If it's unsatisfactory to say "it's just a white gem" there is still one candidate left over. In the Silmarillion we learn about the Noldors' discovery of gems, and in particular Feanor's making of artificial gems:
...he it was who, first of the Noldor, discovered how gems greater and brighter than those of the earth might be made with skill. The first gems that Feanor made were white and colourless, but being set under starlight they would blaze with blue and silver fires brighter than Helluin...
If we really must seek an explanation for what the Arkenstone is in Tolkien's writings, then such a gem may provide this explanation, and it need not even be one made by Feanor or any of the Noldor: a gem "made with skill" that is "greater and brighter than those of the earth" could just as well have been made by the Dwarves (see "cut and fashioned by the Dwarves"), the children of Aule.
Update - 10th March 2015
In History of Middle-earth 11, commentary to the Grey Annals paragraph 22, Christopher Tolkien notes the following from a earlier version of the text ("GA 1" is the earlier version, "them"/"they" is the Dwarves, "Lorien" is the region in Valinor, not Middle-earth, and "Enfeng" is an earlier name for the Longbeards):
The conclusion of this paragraph is wholly different in GA 1:
For Melian taught them much wisdom (which also they were eager to get), and she gave to them also the great jewel which alone she had brought out of Valinor, work of Feanor, [struck out but then ticked as if to stand: for he gave many such to the folk of Lorien.] A white gem it was that gathered the starlight and sent it forth in blue fires; and the Enfeng prized it above a mountain of wealth.
Christopher Tolkien then goes on to note that this idea was subsequently rejected owing to chronological problems: Melian had left Valinor more than 100 Valian Years before Feanor was born, and so could not have taken one of his jewels out of Valinor. In the following version it was substituted for the story of the pearl Nimphelos.
However the choice of words here is hardly accidental, and this, then, is a candidate for the Arkenstone (accepting the difference in the colour of the light).
Christopher Tolkien does not touch on this possibility in his commentary, however.
Assuming the Arkenstone isn't a Silmaril (which is a big assumption not totally dismissed by either of your citations), then we can look for clues in the nature of The Hobbit and of Tolkien's Dwarves.
The Hobbit poses many challenges for people trying to reconcile all Middle-Earth lore into a consistent, continuous whole: it's quite obvious that Tolkien changed his mind about things like goblins/orcs, and the nature of the One Ring, between The Hobbit and The Lord of the Rings, so we can't expect everything in the one book to fit neatly with the other.
So if the Arkenstone is a Silmaril--or anything else within the larger context of Middle-Earth history--it's probably only so as a retroactive declaration after the book was written.
Instead, let's look at what the Arkenstone could be within the context of The Hobbit alone. It's a holy relic of the Dwarves which their tribes rallied around. It glows with its own light. The Dwarves lost it when they were driven out of their homeland by an enemy. Reclaiming it would be a political and spiritual victory of incalculable worth, capable of re-uniting all the scattered tribes of their people.
The language of the Dwarves is Semitic in construction, and Tolkien himself talked about the parallels between the dwarves and the Hebrews of our real-world history. We know that Tolkien began with the languages and then built a world and stories to explain the languages he made; this means the Dwarf/Hebrew connection was in his mind from the very beginning, while the Silmarils may not have been.
In that context, given what we know of the Arkenstone and the nature of The Hobbit as a standalone work, the "true" nature of the Arkenstone seems clear: its origin in the world of Middle-Earth, then, becomes less important to the story than its origin as an allegorical object drawing on real-world sources (just as many of Tolkien's other story elements --especially in The Hobbit-- are directly drawn from real-world mythos). At this point it could still be a Silmaril, and this meaning would take precedent:
The ARKenstone is the Ark of the Covenant.
The Arkenstone is a Simmaril...but not in a "cannonical" sense. As an author, Tolkien conceived the idea of a jewel that shined of its own light. He wrote of it before 1920, but did not publish it...mostly (I think), because he felt he had a larger story to tell and had not perfected it (if this is not understood, read his "Leaf, by Niggle"). Later, he wrote a book for his adolecent children...a simpler one, and easier to publish..."The Hobbit". At that time, I suspect he lacked confidence in his ability to complete "The Simmarrillion" (or at least that it would be published), but wanted to bring certain literary images into a book that would be published. One of them was a jewel that shined of its own light. He still had the concept of the two trees and the "magic sun" in mind, but he was hedging his publication bets as an author. As commented above, "The Hobbit is something of a stand-alone work (including giants throwing snowballs against the mountain) - even Tolkien himself rewrote the "Riddles in the Dark" chapter after publication of LOTR to make the ring into The Ring in future editions. His lack of confidence in the publication of "The Simarrillion" prevented him from taking a similar treatment with the Arkenstone (or Dwarf Magic, or a number of other matters). One must realize that he was not attempting to write the Bible...he was writing fiction for money and for the enjo yment of others, especially in the case of "The Hobbit".
I always took it for granted that the gem must have been another of Aule's attempts at imitating Eru's powers of creation. Just like he created the dwarves. And the stone must have been created by him in the depths of the earth during the first age as he tried to create light of his own.
