We’ve been hunting this one for a long time, but I am reasonably confident (enough to publish) in this answer.
Now, the first two checks against a Stone solution are a) can this be constructed by the original builders of the Tower, and b) was this broken during the events of the game featuring the Tower (in this case, Skyrim). The High King, Skyrim’s political state, and other Nord-centric ideas for the Stone were ruled out by the first, and the Falmer, the Dragons, and other ancient-era thoughts were ruled out by the second.
Furthermore, let us review the Towers and Stones we know:
Ada-Mantia: Convention, the Impossipoint. Enabled.
Red Mountain: Heart of Lorkhan. Disabled (Morrowind).
Crystal-Like-Law: (A person) Disabled (offscreen).
Green-Sap: (A fruit) ?
Khajiit: Mane. Disabled (offscreen).
Orichalc: (A sword) Disabled (offscreen).
(anything in the Marsh?, ?, ?)
Walk-Brass: Heart of Lorkhan and/or the Mantella, both Disabled (Daggerfall).
White-Gold Tower: Chim-el Adabal, the Amulet of Kings. Disabled (Oblivion).
I believe this to be a largely comprehensive list, but it is possible I have omissions or errors.
So: only one of these Towers has a concept as a Stone: Ada-Mantia. Originally enacted by the gods, and reinforced by Talos, Ada-Mantia’s Stone is the asymptotic event that can without doubt be called TIME ZERO. The rest, as far as we know, utilize physical objects with mythic significance as their stones. However, Ada-Mantia and Red Mountain are divinely constructed, whereas the others are all crafted by Elves. Therefore, I see little issue with the idea that, as the Elven Towers are mimicking the Divine, so too could their Stones: a Conceptual Stone is within the realm of possibility, if only barely.
We know the Elven Towers were constructed in mimicry of Ada-Mantia so that the Elves might ascend and return to Aetherius as did their god Auri-El. We also know that this did not work, and that the Towers in fact served to further reinforce the Mundus that they view as their prison. A common question is “Why did the Elves who hate Mundus build Towers to reinforce it?”, and this bears a striking resemblance to a piece of more modern philosophy (anything past Erasmus is modern to me) by Jung: The story of the Cave.
In Jung’s Cave, a man reputed to be a sorcerer was in a cave, and he decided that to escape this cave, he would draw diagrams on the wall. (Mundus’ parallel: Auri-El was a sorcerer in a cave, and he decided that to escape this cave, he would use a Tower.) He drew by a process, beginning with an encompassing piece and refining his process inside it as he went. When he’d finished, his students in the cave with him looked at his finished drawings and said, “Aha! We can use these!” So they drew the diagrams seeking the end result already given, rather than using the old man’s exploratory process, and in doing so they arrived at the end by an exactly backward path. The parallel in Mundus is this: the Elves saw the End: Tower Ascension, and built directly towards it, rather than following the Path. And in so doing, the Towers became inverted and dug down into Mundus and reinforced it, rather than reaching outwards and allowing the Elves to leave.
While we don’t know the exact timeline of Tower construction, it’s safe to assume that they all went up during the same relative period of the Merethic Era. Given the Jungian-Cave state of the Elven Towers, I believe that the Stone of Snow-Throat is the Cave-Method of Tower Construction. It exists so long as the other Elven Towers do.
Now, how did Skyrim’s Main Quest break this? The LDB chased the end result (Alduin banishment) of the Ancient Nord Heroes, but he used a twisted and altered version of their Path. The mechanics were the same but the process was vastly different. He did not explore, he followed, and in so doing made mockery of the action. His eventual defeat of Alduin was both temporary and against the grain of the original victory, and the close association with Snow-Throat, the altering of mythic forces by man, and the fact that all other Elven (Cave-built) Towers of which we know have toppled, meant that the Jung-Cave was strained and broken, and Snow-Throat has fallen.
To sum up: The Stone of Snow-Throat is the twist-purpose-building of the Elven Towers. Those Towers are no longer in play, and a replicated twisted-purpose-process was enacted by the LDB concerning events intimately related to Snow-Throat, on Snow-Throat. These two effects in combination broke Snow-Throat.
This is built off of the following dialogue, and I am fully aware that I am extrapolating and could be horribly wrong.
Oct 15 01:46:02
<+myrrlyn>What is Snow-Throat's Stone Oct 15 01:46:16
%Void_Ghost is Michael Kirkbride.
Oct 15 22:49:07
<+OPG>What are the Stones of Crystal-Like-Law, Falinesti, and Orichalc? Oct 15 22:49:52
<%MK>CLL: A person. F: a fruit. O: a sword.