Only in two games did Link have family (wasn't a complete orphan): alttp and WW (though grandma in WW is NO help at all) so you may wish to Alttp for some answers for a few 'Links'
Perhaps "Link son of Smith" would work for aLttP, since his parents were great swordsman. Often people in medieval times may have been addressed by their passed duties/roles/families, such as Arther from the House of ____. Though in that sense perhaps Link of Hyrule, or the House of Hyrule, or better yet the Champion of Hyrule would be more reasonable as a proper address (such as Cuchulain of Muirthemne, or Cuchulain the Champion of Ireland).
Uncle: "Your parents were Knights of Hyrule... But they stumbled into the Dark World... You will never be able to enter the Dark World and find your parents! [...] They dwell now in the Dark World... After they disappeared, your uncle adopted you and shielded you from the intrigues of the Royal Family..." (A Link to the Past (Nintendo Power), pg. 14)
You've got the idea I had, but I think it would depend on the game. For example, in TWW it would be Link of Outset, while in TP it would Link of Ordon. He shouldn't really have a last name. He is supposed to be the "link" between the player and the game, after all. Let's go with this: Link's last name is the last name of whoever is playing the game.
In many titles, they never mention his parents. In these games, I tend to picture him as an orphan who doesn't know his last name, raising himself. But that's me.