Dawn

From A Wiki of Ice and Fire

Dawn is the ancestral greatsword of House Dayne. It is said to be made from metal forged from the heart of a fallen star. Its blade is as pale as milkglass. The sword is not necessarily passed to the lord of the house, but to the knight considered most worthy to wield it, who is then called the Sword of the Morning. The name of the Daynes' castle, Starfall, and their arms, featuring a white sword and falling star, both reference Dawn.