Kibil-nâla was the Dwarvish name for the river that ran through Lórien, known to the Elves as Celebrant and to Men as the Silverlode. It rose in the mysterious lake known as the Mirrormere (or, in Dwarvish, Kheled-zâram - hence Gimli's words quoted above).
Kibil is one of several words that the Dwarves used for silver (zirak in the name Zirakzigil is also a reference to that metal). The meaning of the Khuzdul word nâla is not known for sure, but Tolkien suggested that it might mean 'course' or 'path', a meaning it would share with Elvish rant. The river's name seems to come originally from the workings of the Dwarves: the notes in volume 7 of The History of Middle-earth suggest that they were the ones responsible for discovering silver in the river.