Ok so since it's Wild's name that is Link Link (according to a reblog), here my headcanons for their name :
Twilight isn't showed to have parents and I don't think he would have inherited a surname from a biological family member or anyone he's not related to since he's also show quite independent when he's still a teen or at best a very young adult. I also firmly believe Hyrule isn't a era where everything is documented like ours and so the surname here have equal chances to be a nickname that stuck or to be inherited from you parent.
But I digress what I mean is that Twilight had his own nickname that became his surname (and his not alone in this). So most notable thing from him at a young age would probably be his bound with Epona and so he was nickname "Little Horseman" which latter give him the complete name of Link Horseman (before that he introduced himself as Link of Ordon).
Aryll have a telescope and her watchtower where Wind must also hang out because that's where we see him sleeping at the start of the game. So maybe those are family heirloom ? Maybe his family are mostly watchmen (or watchwomen since it's seems that Aryll will be the one taking the role after her grandmother if it's true). And if you're wondering why would warchwomen would be needed in a peaceful I would say weather, mail and in the rare case that a boat come mostly. Anyway I headcanon that Wind full name to be Link Clearview.
Hyrule didn't tell anyone his surname when he came to Hyrule but it didn't really matter because everyone started called him "Link the Hero" which eventually evolved to Link Thero.
Time didn't have a surname either, mostly because Kokiri don't really have family. So the times were someone asked his surname he was just confused. Then people would ask "Well where do you come from then ?" and upon answering "Kokiri Forest" people call him "Link Kokiri" or "Link of Kokiri". It happened a few time and he don't know how to feel about that. His emotional dilemma was put to rest when he married Malon and took her name. Now he's Link Lonlon.
Sky don't have parent showed either but I can imagine too many nickname to be throw at him for 1 to stuck particulary. If he must tell a last name he'll probably go by "Link Goodrider" or "Link Bignapper" or something but it's not gonna be stablr, those are just one time, temporary surname until he gets attributed one or just decide on one. Or you know... Get married ;)
Legend would inherit his family name from his uncle so I don't really have a specific idea here. Hum "Link Nobleheart" maybe ?
For Four I'm not going to get very creative : "Link Smithson" is the familly name I see the most for him and I'm not gonna argue with that. I also headcanon that it's an inherited name which mean his Great-grandparents would have named their child "Smith Smithson" for laughs and I love it (I don't know if 'Smith' is the canon name for Four's grandpa but I decided it would be in my fanon)
Ironically Warriors has the same process for his surname that Hyrule has, he's also Link Thero for many people, regardless of if he had already a family name before that. That's the reason the Chain just don't use their family name to call each other. If they're gonna confuse 2 of them anyway what's the point ?
Also since Wild full name is Link Link a lot of them didn't want Wild to be the only one to be called "Link". Plus that would have been weird for everyone.
