Gilderien the Wise was a Prince of House Miolandra, wielder of the White Flame of Vándil and the guardian of Ellesméra since Du Fyrn Skulblaka. No one was allowed to enter Ellesméra without his permission. He appeared to have an insubstantial "avatar", because he appeared to everyone who attempted to enter and disappeared when they passed.
When Eragon and Saphira again visited Ellesméra in Brisingr, Eragon asked Gilderien with his mind if he could pass.
Also in Brisingr, Oromis noted that Gilderien was annoyed with Eragon for waking the Menoa Tree. Not much else is known about Gilderien, although the elf who delivered breakfast to Eragon and Saphira's quarters in Eldest identified himself as "Bellaen du Hljodr" of House Miolandra, the same as Gilderien's House.
His name might come from that of St. Gildas, who was also known as Gildas the Wise.