Kibil is a word that the Dwarves used for silver, the metal rather than the color. The meaning of the Khuzdul word nâla is unknown, but Tolkien suggested that it might mean "course" or "path", a meaning it would share with Sindarin rant. The river's name seems to come originally from the workings of the Dwarves: notes in The Treason of Isengard suggest that they discovered silver in the river.
Other versions of the legendarium
In The Peoples of Middle-earth, in the earliest drafts of Appendix A, the river's name was given as Zigil-nâd. Considering the meaning of zigil, Edward Kloczko proposed that Kibil-nâla is not the name of the actual river, but only of "the reappearance of the river". He connects kibil to Adûnaic khibil, "spring". This seems unlikely, as in The Treason of Isengard (page 241), Tolkien explicitly stated: "Silverlode dwarfish [sic] Kibilnâla".