I don't think that anyone is truely related to each other inside the fence. I think, a select few Americans who are born are sent inside the fence to grow up. Seeing as there is only about one hospital inside the fence, I'd say the children inside the fence are in no way related to their parents. When the children are sent in, I think that they are given names by the government, much like Amanda's new name, Edith. They probably made Tris's last name Prior because they needed one person to unravel everything. They needed one person who everyone sort-of looked up to. Since Edith/Amanda was the one in the video, I predict that in the third book, people will look to Tris more than Tori, who as seen in the book Insurgent, is emotionally unstable because Erudite killed her brother. Tris will, in a way, become leader not just of Dauntless, but of everyone left. Amity, Condor, Dauntless, and Abnegation.
I personally disagree with the theory because Tris and Caleb are clearly related to Andrew and Natalie given the description Tris has made of herself and her brother. The theory also crumbles because Tobias has exactly the same eye color that Marcus has, therefore they are related.
Before the attack on Abnegation, The Abnegation leaders were about to release the video shown at the end of Insurgent, which tells everybody they need to go outside the fence in order to restore society. Jeanine didn't want that to happen for one reason or another. I'm not sure if we are told exactly why, possibly because she knows what's out there is bad, or she's just power hungry, but she wants it bad enough that she sent the attack.
Hope this helps.
I think that Edith Prior AKA Amanda moved in to a selection of people. Those people, now isolated from the other genetically damaged or whatever people, bond with each other so obviously they can fall in love. I think Edith is actually related to Beatrice.
I am pretty sure that there is only 2 generations which makes Edith Prior Tri's grandmother. Which makes sense becuase Edith looked to be a dauntless leader and Natalie Prior was dauntless born. and as said before you can keep your surname and or change it.