This is one of the items that can be in your inventory to obtain the Eye of the North quest Fire and Pain. You must have previously completed The Missing Vanguard.
If Gwen's Broken Flute is in your inventory when you first meet Gwen in the Hall of Monuments, she will say "I appreciate that you kept these relics of my past, but we must put them aside for now." Then she continues the normal dialog about the Ebon Vanguard. -Wiki
Says you can use it for the quest, but you need to have completed The Missing Vanguard first :)
"People are like slinkies; Basically useless, yet so amusing to watch fall down the stairs."
"No! He would kill you like a small dog. Let your anger be as a monkey in a piņata... hiding amongst the candy... hoping the kids don't break through with the stick!" - Master Tang
