Link is an Orphaned Hylian, meaning whatever his mother's last name was, was his. He was adopted by the Kokiri and the Great Deku Tree. Nothing else is really discovered about his past origin or family.
nobody in the Zelda series has a last name
I used to think when I first got the Elixir Soup, his grandmother named it after their last name, so I thought Elixir was Link's and Aryll's last name since I didn't know what Elixir meant.. XD