I believe that the movie has nothing to do with time travel and, as Oliver_C mentioned in a comment, the butterfly effect is not an ability, it is just cause-effect gone to extremes ("Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?").
The following lines are solely based on my own interpretation of the movie, hence no external links or proof will be provided. It is no need to say that spoilers will follow.
The whole movie is about the tough decision a child is facing when having to decide whether to live with his mother or with his father. It is not easy to choose something like this, especially when you're a child. This is about 90% of the movie: the child is actually standing between his parents trying very hard to make the best decision. In his mind, there are a lot of possibilities. THIS is what we actually see: it is not time travel, nor seeing the future, but the child's imagination. He tries to picture his whole life with each of the parents in order to be able to choose whatever should seem better.
We cannot be sure whether one of the stories actually occur. Absolutely nothing is certain - everything is pure imagination. It is like day-dreaming: you picture something and then one thing leads to another and you find yourself wondering what's it like to be old and having grandchildren. Still, in the movie, this "daydreaming" part is not as pleasant as it should be, it happens under a lot of stress and all this thoughts probably occur in just a few minutes.
To my mind, this is the movie's theme: the toughest decision a child has to make - choosing only one of his parents, that is deciding his own future.
In the end, after being asked which story is the true one, the old man refuses to pick one. He states that all of them are equally true and that is no right, nor wrong answer. This has multiple meanings.
In the first place, I don't think his intention was to actually tell the story of his life. When you are the eldest person in the world and dying, I don't believe that your main intention is to tell the people how your life has been. It seems a better decision to tell the whole world a message. After so many years of experience, the question shouldn't be "What have you done so far?" but "What have you learned? What would you tell to all the young people?". Nobody should actually care about the ups and downs in another person's life. The title is Mr Nobody, meaning that he was no extraordinary person, he was just nobody, like everybody else. Returning to the original question and it's interesting answer, I believe that the message is the following: we shouldn't consume ourselves trying to figure out what is the correct decision in life. Life is about living, one thing will lead to the other and you cannot predict what will follow. As you have seen in the movie, apparently bad decisions have led to happy moments, while doing always the right thing might get you bored, dissatisfied and maybe finally killed. That is why there is no right, nor wrong decision in life. They are all just options.
In the second place, there might be a reference to quantum mechanics. You can read more about various theories here about the many-worlds interpretation and here about the Copenhagen interpretation. The very popular experiment with Schroedinger's cat might also be a good example. To sum the theories up: when facing multiple possibilities, before making a choice/some sort of measurement, every possible outcome is actually an alternate universe. Now, one theory states that every time a decision/measurement is made, many other parallel universes are created. This way every possibility actually gets materialized, only it happens in parallel universes. Other theories state that upon deciding/measuring results, all parallel universes collapse into a single one, but before measurement everything is equally real. Hopefully I haven't mixed them up and understood this ideas good enough.
All in all, it is my opinion that all the tiny actions with unexpected huge reactions are not necessarily true. Everything is narrated by the main character and he couldn't have known what was happening in other parts of the world (see the episode with the guy working at a shoe factory being fired). What is more, all the possible lives are made of various decisions that have interesting effects in time. All these are references to the butterfly effect but, as I have said earlier, nothing is necessarily true. Everything is possible.