I'm not clear on how that happens. Does it "cover up " the sign you would be born under?

Birthsigns no longer exist in Skyrim; instead your character choses a standing stone that will give him/her blessings. Something of the kind existed in Oblivion.

http://www.uesp.net/wiki/Skyrim:Standing_Stone

Nevertheless, signs are still organised like this: http://www.uesp.net/wiki/Lore:Birthsigns#Constellations

The Serpent doesn't have a Guardian Stone.

Mabie it's the TES equivalent of a leap year birthday.

Not entirely correct, a book describes that the standing stones are only usable by a lucky few, most people are still born under a sign. No idea what the book is called though

And what is an un-star which the serpent is said to consist of.

Wait, so people who can use them aren't born under any sign?

http://imperial-library.info/content/watcher-stones

And I don't know, the guy in the book was born under a sign and believed that he could become worthy of using the stones. Perhaps that is true, perhaps you can become worthy, perhaps you have to be blessed by the gods, like the dragonborn is or perhaps you have to be born without a sign.

Okay. Do we know how people born under signs would be born under the Serpent though? Not seeing the answer on UESP.
Did a bit more digging. Didn't find anything official, but everything suggests that if you were about to be born under a sign and the Serpent attacked it you would be born under the Serpent instead. That makes sense.

The Serpent does have a stone in Skyrim.

Yeah, on an iceberg north of Windhelm.

I'm not talking about a stone, I'm talking about a guardian stone. The serpent doesn't belong to the warrior guardian sign, neither to the mage guardian sign, nor to the thief guardian sign. You would have understood what I was saying if you had clicked on the second link I posted and actually looked what was said on this page. Here: http://www.uesp.net/wiki/Lore:Birthsigns#Constellations

You mean it dosen't have a guardian constellation? Everyone knows that. That's the point of the Serpent.

Your post was ambiguous and ultimately irrelevant if that was what you were trying to say. Perhaps better to say that the Serpent does not fall under the three Guardians, or that it does not recognize a Guardian sign. Something that most of us understand already, but a legitimate clarification nonetheless. You spoke of stones for some reason, the Guardian Stones being irrelevant, indeed, to all others in Skyrim. Only three constellations have Guardian Stones, those of the three Guardians, and none affects any other stone or constellation to a great degree.

I can't quite remember where I see it, but I do have seen some theory in this forum that when the moving Serpent interferences into another constellation, to some degree people born at that time would be considered born under Serpent instead of the constellation interferenced.

Basically what I was saying, only you said it in a way that makes sense.

[speculation] The reason why the standing stones of Cyrodiil and Skyrim do not appear to work to most mortals, yet unfailingly yeild their powers (except for some of the doomstones, which have a fame requirement) to player characters is because they are purpose built to only work for the various player characters. They were not shrines, waypoints, or other things, but purpose-built tools designed to arm the prophesied Champions of old Tamriel. [/speculation]