The White Flame of Vándil was a mysterious weapon, only mentioned twice, in both Eldest and Brisingr. Devised by the elf of the same name, it was wielded by Gilderien the Wise, in defense of Ellesméra. Hardly anything was revealed about what it was exactly or how it worked. It is described in Eldest as follows:
In late afternoon, the gloom shifted to reveal an elf standing before them, sheathed in a brilliant ray of light that slanted down from the ceiling. He was garbed in flowing robes with a circlet of silver upon his brow.
The "brilliant ray of light" probably refers to it, as Arya reveals later in the chapter.