From A Wiki of Ice and Fire
The name "Ice" is a legacy from the Age of Heroes, and predates the current sword. About four hundred years before Robert Baratheon's reign, the Valyrian steel greatsword was spell-forged in Valyria and acquired by the Starks, who named it after that legacy. It became the sword of the Lord of Winterfell. Eddard Stark used the sword to personally execute condemned prisoners.
A Game of Thrones
Lord Eddard Stark uses Ice to execute Gared, a deserter from the Night's Watch. After Eddard is arrested for treason, the royal executioner, Ser Ilyn Payne, is given the sword and uses it to execute Eddard himself at the Great Sept of Baelor.
A Storm of Swords
|“||The blade was Valyrian steel, spell-forged and dark as smoke. Nothing held an edge like Valyrian steel.||”|
|“||What has Ser Ilyn done with my father's sword?||”|