The name Ice is a legacy from The Age of Heroe's, and predates the current sword.About four hundred year's 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 then became the sword of The Lord of Winterfell.
Eddard Stark used the greatsword in battle and to personally execute condemned prisoners. After his capture, Sir llyn Panye was given the sword and used it to execute Eddard himself. llyn carried the blade for a short while untill Lord Tywin Lannister took it from him. Tywin then commisioned Tobho Mott to melt down Ice and had it reforged into two swords, Widow's Wail and Oathkeeper.
Widow's Wail was given as a wedding gift to his grandson King Joffrey, the first of his name who died at his wedding feast. It was then passed down to his brother Tommen I.
The second sword was given to Sir Jamie Lannister, The Kingslayer, who then gave it to Brienne of Tarth who named it Oathkeeper when Jamie charged her in finding Sansa Stark.