Been there, done that. In my story the name of the president of the US changes. And of course it's a deep reference to quantum mechanics and has absolutely nothing to do with not being able to remember the names of my own characters.
More seriously if you want to pull this off then make sure that the reader gets that you do it intentionally. If its just things changing, people will think it's a mistake. If they get it's intentional but random and unimportant, it seems a bit silly and pointless. To make it a fun part of the story it must be somehow predictable. Why not invent your own “physics” according to which this works?
Is it Anathem? In particular, this detail seems to match what you remember:
Late in the novel, a thousander named Fraa Jad tries a keypad and hits buttons at random. The door opens, and the implication is that Jad has willed himself into that universe where the random keys were the right keys.
There's aliens, parallel universes, metaphysical themes, and an Earth-like planet. The only detail that doesn't seem to match is the aliens putting humans in reeducation camps but the "humans" of this planet do put some of their own people in camps to think about philosophy.
Even if one is willing to buy many worlds theory (most physicists are not) the differences between worlds are limited to QM uncertainties. The only way in which you get macroscopically different worlds is by butterfly style amplification of uncertainty.
I.e. A quantum fluctuation gives an electron a tiny bit more energy. This causes an atom to be ionized, which causes a protein to change its shape, which causes a vesicle to open which causes a neuron to fire, which sets of a cascade of neurons firing in your brain, which causes a change in large scale brain states which causes you to buy the red shirt and not the blue one. So later you have a red shirt and not a blue one. (which very likely matters not at all for the rest of the story)
I don't know what I was expecting when I clicked on this page, but man, this got deep.
YES!