Shhh! The language of the Dwarves is “a secret which unlike the Elves, they [do] not willingly unlock, even to their friends”. Aside from the names of a few places (which Gimli let slip to his Companions) and the famous Dwarven battle-cry “Khazâd! Khazâd!” (which has been heard by many, though sometimes last), the Dwarven tongue is a “treasure of the past” which has been guarded and tended by this “race apart”.
The name Gimli, like that of all Dwarves, is actually Mannish in origin, a custom adopted in the Third Age when the Dwarven diaspora spread out from their lost ancestral homes to live and work among the Men of Middle-earth. Like his Dwarf names from The Hobbit, Tolkien pulled the name Gimli from Old Norse epic poetry and says in Letter #297 that the name probably means ‘fire’.
Appendix F tells us, however, that Gimli and his kin have “their own secret and ‘inner’ names, their true names”. Whatever Gimli’s “true name” was, Tolkien never wrote about it and, presumably, even Legolas never learned it. He took his secret with him to Valinor. As for other Dwarves, they too “never revealed [their true names] to any one of alien race. Not even on their tombs do they inscribe them.”
Wow, after all these years? I feel like I hardly know this Dwarf when I play him on the table anymore…