I too have been lurking and looking at this thread

And Googling to try to find out the history of Corncockle quarry and whether it was ever known by any other name - including just a generic name. No luck.
I say a 'generic name' because I wondered from the start if it might be Flagg referring to a quarry where
flags/flagstones were obtained - the 'ff' at the beginning being the old-fashioned way of representing an upper case 'F' and the double g just a quirky spelling of Flag.
Googling finds that Corncockle quarry
does produce flagstones (i.e. paving stones) e.g.
"The sandstones of the
Permian or Lower New Red are usually of red and purplish tints, rather open and soft in texture, and of no great durability. When carefully selected, however, both building-blocks and flagstones can be obtained from the formation, as may be seen at Lochar Briggs and Corncockle ..."
The only thing I could find about its history was that it must have been closed for a time (but for how long?) because there's a reference to Sir William Jardine having access to Corncockle Quarry on his estate; apparently he was fascinated by fossil tracks on the flat surfaces of the beds. And it says that he re-opened the quarry in 1847.
If it was referring to a quarry where flagstones were quarried, then perhaps it could mean Corncockle itself or perhaps there was a minor outcrop nearby which was also quarried for flagstones but in a smaller way ...
Just a thought.
Of course, it could be a name - but FLAGG seems to be a fairly uncommon name and virtually unknown in Scotland.
JAP