In these extracts you can read the origin for Narya's name

Three Rings for the Elven-kings under the sky, Seven for the Dwarf-lords in their halls of stone, Nine for the Mortal Men doomed to die, One for the Dark Lord on his dark throne, In the land of Mordor where the Shadows lie. One Ring to rule them all. One Ring to find them, One Ring to bring them all and in the darkness bind them, In the land of Mordor where the Shadows lie. "The Lord of the Rings"
Now these were the Three that had last been made, and they possessed the greatest powers. Narya, Nenya and Vilya, they were named, the Rings of Fire, and of Water, and of Air "The Silmarillion"
For this is the Ring of Fire, and herewith, maybe, thou shalt rekindle hearts to the valour of old in a world that grows chill. "TheSilmarillion"