Normally, you can assume your thought processes are uncorrelated with whats out there. Newcomb-like problems however, do have the state of the outside universe correlated with your actual thoughts, and this is what throws people off.
If you are unsure if the state of the universe is X or Y (say with p = 1/2 for simplicity), and we can chose either option A or B, we can calculate the expected utility of choosing A vs B by taking 1/2u(A,X)+1/2u(A,Y) and comparing it to 1/2u(B,X)+1/2u(B,Y).
In a newcomb-like problem, where the state of the experiment is actually dependent on your choice, the expected utility comparison should now be ~1u(A,X)+~0u(A,Y) vs ~0u(B,X)+~1u(B,Y).
In this case, it boils down to "Is u(A,X) > u(B,Y)?".
It is not enough for Omega to have a decent record of getting it right, since you could probably do pretty well by reading peoples comments and guessing based on that.
If Omega made its prediction solely based on a comment you made on LessWrong, you should expect that if you choose A the universe will be in the same state as if you choose b- knowing your ultimate decision doesn't tell you anything, since the only relevant evidence is what you said a month ago.
If, however, Omega actually simulates your thought process in sufficient detail to know for sure which choice you made, knowing that you ultimately decide to pick A is strong evidence that omega has set up X, and if you choose B, you better expect to see Y.
The reason that the answer changes is that the state of the box actually does depend on the thoughts themselves- it's just that you thought the same thoughts when omega was simulating you before filling the boxes/flipping the coin.
If you aren't sure whether you're just Omega's simulation, you better one box/pay omega. If we're talking about a wannabe Omega that just makes decent predictions based off comments, then you defect (though if you actually expect a situation like this to come up, you argue that you won't)
Omega's actions depend only on your decision (action), or in this case counterfactual decision, not on your thoughts or the algorithm you use to reach the decision. The action of course depends on your thoughts, but that's the usual case. You may move several steps back, seeking the ultimate cause, but that's pretty futile.
Related to: Can Counterfactuals Be True?, Newcomb's Problem and Regret of Rationality.
Imagine that one day, Omega comes to you and says that it has just tossed a fair coin, and given that the coin came up tails, it decided to ask you to give it $100. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don't want to give up your $100. But see, Omega tells you that if the coin came up heads instead of tails, it'd give you $10000, but only if you'd agree to give it $100 if the coin came up tails.
Omega can predict your decision in case it asked you to give it $100, even if that hasn't actually happened, it can compute the counterfactual truth. Omega is also known to be absolutely honest and trustworthy, no word-twisting, so the facts are really as it says, it really tossed a coin and really would've given you $10000.
From your current position, it seems absurd to give up your $100. Nothing good happens if you do that, the coin has already landed tails up, you'll never see the counterfactual $10000. But look at this situation from your point of view before Omega tossed the coin. There, you have two possible branches ahead of you, of equal probability. On one branch, you are asked to part with $100, and on the other branch, you are conditionally given $10000. If you decide to keep $100, the expected gain from this decision is $0: there is no exchange of money, you don't give Omega anything on the first branch, and as a result Omega doesn't give you anything on the second branch. If you decide to give $100 on the first branch, then Omega gives you $10000 on the second branch, so the expected gain from this decision is
-$100 * 0.5 + $10000 * 0.5 = $4950
So, this straightforward calculation tells that you ought to give up your $100. It looks like a good idea before the coin toss, but it starts to look like a bad idea after the coin came up tails. Had you known about the deal in advance, one possible course of action would be to set up a precommitment. You contract a third party, agreeing that you'll lose $1000 if you don't give $100 to Omega, in case it asks for that. In this case, you leave yourself no other choice.
But in this game, explicit precommitment is not an option: you didn't know about Omega's little game until the coin was already tossed and the outcome of the toss was given to you. The only thing that stands between Omega and your 100$ is your ritual of cognition. And so I ask you all: is the decision to give up $100 when you have no real benefit from it, only counterfactual benefit, an example of winning?
P.S. Let's assume that the coin is deterministic, that in the overwhelming measure of the MWI worlds it gives the same outcome. You don't care about a fraction that sees a different result, in all reality the result is that Omega won't even consider giving you $10000, it only asks for your $100. Also, the deal is unique, you won't see Omega ever again.