So, there seems to be some debate on what exactly is going on with the whole "The leader of the Institute is apparently your son, Shaun" point in the storyline. What I am going to do is put forward all the information I've gathered on the subject, and then put in my opinion/theory. Then I would like to see what you all think.
So in the main storyline, you are introduced to man who goes by Father, and claims to be the Sole Survivor's son, Shaun. According to father, it has been 60 years since he was taken from Vault 111 by Kellogg, and that he has ascended to be the leader of the Institute. To show that he is grateful for the Sole Survivor's role in being his parent, he gives the player unrestricted access to the facilities of the Institute. However, many people question whether he is truly Shaun. While there are a few avenues that the Sole Survivor could take to confirm or deny this, like a blood test or checking possible databases of detected openings of Vault 111, none of them are available. This is for two reasons, one more solid than the other. On one hand, the Institute could fake or tamper with almost anything the Sole Survivor could attempt to do, twisting it in their favor. On the other hand, any avenue for the truth could turn the player's opinions too quickly, which leaves the answer very obvious and one-sided. Bethesda most likely wanted this important part of the story to be left to player interpretation, making the storyline more interesting. So in order to find the truth, one must look to the evidence that is displayed throughout the game.
First of all, why would the Institute only take a child from Vault 111? The best possible explanation is that with a child, they could continue to observe how the body functions throughout growth, thus being able to perfect their gen-3 synths of any age. But why would they let a child, whose only purpose was being a resource, ascend to any rank of Institute scientist? This is most likely for two reasons. One, a large portion of Institute scientists are not 'evil' and are seemingly kept in the dark about the Institute's darker side (Like Virgil and his FEV research later on) so they likely grew attached to the child shaun during the time he was a resource, and would be adamant against letting Shaun be simply disposed of. Two, a child fresh from a Vault would have no past experience with things like morals, making that child perfect for molding with the more malicious side of the Institute.
But there are many inconsistencies to keep in mind. One, look at Kellogg at the time of him taking Shaun from the Vault and the time of the Sole Survivor catching up with him. No difference in appearance that would be caused by age. While it is safe to reason that his life has been prolonged by the Institute, it makes no sense that only Kellogg would get this and not Father. If you are going to keep your best weapon (Kellogg) in good condition, why not keep yourself in good condition too? The Institute isn't stupid like that, if they had that technology they would use it to the best of their abilities. Two, when you finally encounter Kellogg and he tells that Shaun is with the Institute, he mentions that he's a good kid. While it is feasible for Kellogg to view even the aged Father as being younger than him if the age extending point was true, he simply wouldn't do that. Say Father is actually Shaun, combine his age by the events of 2877 and the fact he spent a couple hundred years in an icebox, and it's obvious that he would be too old for him ever to be referred as a kid, especially not by Kellogg. Kellogg might be a murderer and a mercenary, but he is honest enough to not lie about that. Besides, it is revealed through his memories that he personally wishes to see the Institute's downfall, so he probably would have told the Sole Survivor if his son grew up to be the head of a murderous corporation he hates, rather than saying that he's a good kid. The final inconsistency is that even though Father greets you as family when you finally meet him, up to that point he has let the Institute's Synths target you like a regular wastelander. If he truly saw you as family, he would have brought you in as soon as possible. He wouldn't risk you dieing at the metal hands of those under his control, putting your blood on his hands. He also doesn't seem to care about the Sole Survivor's spouse dying, but that can be assigned to the no previous morals point.
The way I see it, Father is indeed lying to you, but on two fronts rather than the obvious one. He is not your son, Shaun, and seeks to use you as a replacement for Kellogg, seeing as you had the skill to kill him. He wants you to be the Institute's new weapon, and is willing to pretend that he is your son to do so. He is also lying about the child being a synth. The child is the actual Shaun, and was taught to believe otherwise. By time Father realizes his best weapon has turned towards him in one way or another, he turns the Synths back against him for a relatively short moment of cannon-fodder safety. He uses this time to reveal the truth to Shaun, and then purposely gives himself some form of material to make himself sick in a last ditch effort to save himself through the Sole Survivor's compassion. Instead, he realizes too late that he failed to keep himself and the Institute above water, and being the intelligent man he is, accepts his fate. By doing this, he dooms himself to die, and unintentionally gives the true shaun a chance at life.
So, what do you all think about this? What are your theories and opinions?