It’s the family name that lives on.

If another house can seize one of our own, and hold him captive with impunity, it means we’re no longer a house to be feared! Your mother’s dead, before long I’ll be dead, and you and your brother and your sister and all of her children. All of us dead, all of us rotting in the ground. It’s the family name that lives on. It’s all that lives on.

Tywin Lannister, Game of Thrones, Season 1, Episode 7