Okay, here is the inspiration, from Appendix F of LOTR:

But in the Third Age close friendship still was found in many places between Men and Dwarves; and it was according to the nature of the Dwarves that, travelling and labouring and trading about the lands, as they did after the destruction of their ancient mansions, they should use the languages of men among whom they dwelt. Yet in secret (a secret which unlike the Elves, they did not willingly unlock, even to their friends) they used their own strange tongue, changed little by the years; for it had become a tongue of lore rather than a cradle-speech, and they tended it and guarded it as a treasure of the past. Few of other race have succeeded in learning it. In this history it appears only in such place-names as Gimli revealed to his companions; and in the battle-cry which he uttered in the siege of the Hornburg. That at least was not secret, and had been heard on many a field since the world was young. “Baruk ’!Khazâd! Khazâd ai-mênu!” – ‘Axes of the Dwarves! The Dwarves are upon you!” Gimli’s own name, however, and the names of all his kin, are of Northern (Mannish) origin. Their own secret and ‘inner’ names, their true names, the Dwarves have never revealed to any one of alien race. Not even on their tombs do they inscribe them.

So, I am sure I have mentioned that I have no self-restraint a love of extreme detail! I went with this concept to the absolute limit. If we assume that Mahal makes each Dwarf personally and with utter love, I thought that perhaps instead of Dwarves naming their children twice – once with an ‘outer’ name, and once with a ‘secret and sacred’ name, it wasn’t beyond the realms of possibility that Mahal himself had given the Dwarf their sacred name. Mahal would have written it into their very bones, sent it swirling through their blood.

I went a bit further. That sacred name would be the whole and soul of a Dwarf – the very essence of their best self. It would be inextricably linked to a Dwarf’s very personality. (i.e. Gimli’s dark-name is Sansûkhâl [One who sees with perfect/pure sight], which refers to many things. His bluntness, his role in defeating darkness, his overcoming institutionalised racism, his perception of Elves, Dwarves and Men, his caring and comforting nature.)

Aaaaaand a bit further yet. If they are sacred names, then I reasoned that it was likely that Dwarves only shared such names with close kin. Also – they would customarily be shared only under stone, because Khuzdul itself is not to be used in outer lands amongst other peoples. Therefore – “Deep name” or “dark-name.”

And so, the ‘outer name’ (i.e. Gimli) would also be known as a ‘sky-name’ – to be used in the open air.