Pickle, it's not a bunch of descendants that you see in every game. Aside from a couple of connections fans like to make that aren't proven, and the few games that share the same Link, all of the Links are completely unrelated. They're not the same person. They're not reincarnations. They're not descendants or ancestors.
The name Link comes from Nintendo making the character of Link a connection/link to the player.
^ This. This is the original reason for them naming Link this. I know it's a real name, but that's not the real reason they named him it.
Link is a single character designed so that the player has a familiar anchor to the game world. He's basically the same in every game because he's iconic but also because that character doesn't need to change. As of late, they've given him more of a unique personality and in-game storyline connection, but they still always design his character in such a way so that the player has an easy time identifying with Link.
All these things combined are
probably the reason I've never been particularly fond of Link's character and rarely feel like writing fanfiction about him, because he's not exactly a full character in many ways.
Now, should all of this stay the same? I'd say not necessarily. I think "Link" is a good default name, and there's no reason for it to change, but I think toying around more with Link's character and changing how he looks and such isn't a bad idea. I also like the idea of a highly customizable Link, perhaps even with selectable gender. That could be interesting. At the same time, I don't think there's necessarily anything wrong with keeping things exactly how they are. I don't see any problems.