I was wondering; what is the name of the place Gandalf falls to when the Balrog drags him down, after the fight in Moria?
What information from the Legendarium is there available about this place?
"Thither I came at last, to the uttermost foundations of stone". And: "at last he fled into dark tunnels. They were not made by Durin's folk...Far, far below the deepest delvings of the Dwarves, the world is gnawed by nameless things". From The Two Towers, Chapter 5.
They then fight all the way up the "Endless Stair" to where I believe you (and others viewing this question are likely thinking of: the mountaintop where the battle concluded, Zirakzigil.
From the LotR Wikia:
Neither he nor the Balrog was killed by the fall, and Gandalf pursued the creature for eight days until they climbed to the peak of Zirakzigil. Here they fought for two days and nights. In the end, the Balrog was cast down and it broke the mountain-side as it fell. Gandalf himself died following this ordeal and his body lay on the peak while his spirit travelled outside of Time.
There is nothing else about this place, and this was probably the only time it was ever visited by sentient and rational beings. There are suggestions (as quoted above) that there are beings down here, but they are outside the stories of the sentient beings of middle earth, and so are not recorded elsewhere. Gandalf may be the only being that ever went down there and returned to the light.
It is an indication that the created universe is wider than the tales above. And that the makeup of the world is not necessarily as clear as it seems - there are mysteries deeper than anyone understands.
and in no iterations of the text has it had a name other then the "bottom of the world" or the "deep/deepest".
In the original versions of the text, Christopher Tolkien, editor of the History of Middle-earth series, gave us insight into his Father's ideas for what happened in Moria.
Gandalf's story. Overcame Balrog. The gulf was not deep (only a kind of moat and was full of silent water). He followed the channel and got down into the Deeps. ?? Clad himself in Mithril-mail and fought his way out slaying many trolls.
The Treason of Isengard - Chapter XI: The Story Forseen from Moria
We can clearly see how the story in the final version had changed from the original version to the published version (at the bottom).
The above is however only the first notes on the matter, the first draft (written over erased pencil) of Gandalf's story of his escape from Moria is copied below. (When exactly it was written is unknown but CT suspects it was in the course of writing 'The King of the Golden Hall'.
... but tells how he passed through fire (and water?) and came to the 'bottom of the world', and there finally overthrew the Balrog, who fled.
The Treason of Isengard - Chapter XXIV: The White Rider
Tolkien again neglects to name the place and simply calls it the "bottom of the world". Furthermore, in this draft, Tolkien seems to be unsure whether there was indeed water at this occurrence.
A very rough and unfinished draft for the final form and placing of Gandalf's story is found below:
he was a thing of slime, strong as a strangling snake, sleek as ice, pliant as a thong, unbreakable as steel.' Of the 'dark things unguessed' that gnaw the world 'below the deepest delvings of the dwarves' he says: 'Sauron alone may know of them, or one older than he.' And after his words 'I will bring no report to stain the light of day' the text continues:
'...Little had I guessed the abyss that was spanned by Durin's Bridge. Did you not?' said Gimli. 'I could have told you had there been time. No plummet ever found the bottom - indeed none that was ever cast therein was ever recovered.
Christopher goes on to describe it as such:
It is interesting to look back to my father's original ideas about the chasm in the passages referred to in note 4: 'probably fall is not as deep as it seemed... eventually following the subterranean stream in the gulf he found a way out', and 'The gulf was not deep (only a kind of moat and was full of silent water). He followed the channel and got down into the Deeps.'
From the examples that present themselves in the text below and the notes on previous drafts above, there is little to go on besides that there was water, and certainly later a stair that they climbed.
Yet it has a bottom, beyond light and knowledge,’ said Gandalf. ‘Thither I came at last, to the uttermost foundations of stone. He was with me still. His fire was quenched, but now he was a thing of slime, stronger than a strangling snake.
‘We fought far under the living earth, where time is not counted. Ever he clutched me, and ever I hewed him, till at last he fled into dark tunnels. They were not made by Durin’s folk, Gimli son of Glóin. Far, far below the deepest delvings of the Dwarves, the world is gnawed by nameless things. Even Sauron knows them not. They are older than he. Now I have walked there, but I will bring no report to darken the light of day.
The Two Towers - Book III, Chapter 5: The White Rider
From the final version in the novels, we can see that the version found at the end of "The Treason of Isengard" was very close to that in the novels, and little changes were made there.
In the game, Lord of the Rings Online, the area is known as The Foundations of Stone. That is where the Balrog and Gandalf fell. Close by is the endless stairs that leads up to the peak of Ziragzigil. The Endless Stairs are trashed and unusable due to the battle that went up it. The stairs were used by Dwarves during Durin's reign.