SPOILERS: About the Son and the Ending

I think it was a boy that he saw was fighting for life -- the only one to survive the gas chambers -- and felt an immediate love and empathy for him. At that moment, Saul realized that the boy should have been living and not himself. He decided to give him the only gift he could receive -- the possibility of an afterlife.

To be honest, this was the part of the movie -- this idea that he was NOT his son -- that makes me cry when I think of it. A beautiful gesture.

I'm fairly certain that he did have a son with someone other than his wife, who would have been roughly the same age as the boy he "adopts" and whom he has not seen since very young childhood or even infancy. I don't think he makes the emotional connection otherwise; it is too quick and too strong.

I think that what he believes at the time that he first encounters him is that the boy possibly could be his son. That's a thought you would have, literally, as in, I cannot rule this out. I think that as the action continues he naturally begins to more and more treat the boy as if it were true. There's something very deep in deciding consciously to let your unconscious mind take over and do that. I think there is a level where he knows the boy is very likely only symbolically his son, and another, stronger level where he feels as if he absolutely is.

Prepare your minds for a new scale of physical, scientific values, gentlemen.

My take is that this was his actual son. One of reasons this film is so compelling, is that it is telling several stories at the same time; it's incredibly complex. One of the undercurring stories is that Saul had become toitally numb and indifferent to everything happening around him. He was also very guarding of his personal information, because any one of his co-workers would have sold him out and not think twice about it. In fact, immorality and even amorality seems to be the major point of the film. Seeing his son, Saul snapped out of it.

Either way, this film is open to interpretation, and there is nothing conclusive about the story.

''My guess is that he isn't. It's just a young boy that reminds him of either his actual son.''
Yes, this. He was clearly delusional.

''They are shaping me into something gaudy. Something lethal.''

He probably wasn't his son, but he was convinced that he really was. However, by going on a quest to bury his "son", Saul reconnects with his humanity. Like it's been pointed out, Saul wasn't an observant jew, he only needed 10 people to pray the Kaddish, not a rabbi- he only wants to do what is right for his "son", the humane thing, even if it is irrational to everyone else around him. He is trying to make sense of the place that is basically hell on Earth.

The correct answer to this question is: It doesn't matter.

I personally doubt it was his son, but either way, he just needed some mission to his life in hell, to give himself a reason to exist further. That's why it is without relevance in what relationship Saul actually was with that kid

I don´t think it was his son, or even reminded him of his son (after all, in at least two scenes, i think, someone questioned the fact that he even had a son), I think he was in a state of psychosis, loosing his sanity over the harrowing events surrounding him. I think he was grasping for something that was worth living for and to give his son a proper burial and passage into heaven(?? I have to admit that my knowledge of judaism is rather limited), hence the rabbi, was something to give him purpose.

I am Ubik. Before the universe was, I am. I made the suns. I made the worlds.

I think the boy is (figuratively) Florya from Come and See, a Russian WWII movie to which Son of Saul owes a clear stylistic debt. The ending is, I believe, a reference to that movie by way of intersecting narratives. Come and See's protagonist Florya is a character who is both a witness to senseless violence and the beneficiary of senseless good luck in slipping through the cracks to survive a chaotic hell-scene (much like Saul). Watch the ending of Come and See and compare with Son of Saul. The boy even bears a slight resemblance to Florya.