From the words of Roller, In search of god the mother: The cult of Anatolian Cybele (1999), p.68: "Thus citations in Classical literature make it appear likely that the epithet kubileya (or kubeleya) was derived from the name of a natural feature of the landscape, probably a mountain."
After just writing the previous post and then reading this, I find myself crapping my pants right now. Time for the Depends! Hahaha. Towards the New Year, I had already been scanning through Hesychius' Glossai, focussing on words identified by the author as 'Cretan' in particular, and making detailed personal notes. I'm dazzled by the network of vocabulary that's unfolding before my eyes, much of which is mirroring what I've discovered in Etruscan. I really feel more and more that Hesychius' work, as negligent as he was in marking overtly the language of the words he cited, is a treasure trove of hints that may round out our understanding of the Proto-Aegean language family. I suspect that a common Aegean vocabulary can be reconstructed if we're diligent. A reconstructed Proto-Aegean will simultaneously complement our understanding of Indo-European etymology and weed out falsely reconstructed IE roots like *peleḱus 'axe'.
Long story short, In search of god the mother: The cult of Anatolian Cybele shares the fact that Hesychius labelled the gloss Κύβελα as ὄρη Φρυγίας 'mountains of Phrygia'. A short-sighted scholar might assume that the word is thus a native 'Phrygian' word and stop there. However, logically, we must accept that this in itself only suggests to us that the mountains of Phrygia were named this by contemporaneous locals, most of which being presumably Phrygian at the time. In light of my theorizing on Minoan *kapada meaning 'summit', I believe I might be able to answer the questions posed by the author of that book above.
The word may be Phrygian, sure, but more fundamentally, it strongly looks like it's of Minoan origin in light of this fact, several other glosses by Hesychius starting with kub-, and the apparent name for Minoa I spoke just a few days ago. I've therefore refined my reconstructions to *kaupada 'head' (with a secondary sense of 'mountain-peak') and thus *Kaupadaria 'Minoa; Land of the (Two) Peaks'. As for the value of *d here, I might be able, thanks to this investigation, to converge on a value of [θ] ~ [ð]. In fact, if I may be so bold, the very Minoan iconography of twoheadedness itself could have been evoking the plural form *kaupadar 'heads' and thus the name of Crete. I'm getting goosebumps. I hope I'm not wrong.