(this comment assumes "Reward = $1,000,000 * P(onebox)")

You misunderstand frequentist interpretation - sample size is 1 - you either decide yes or decide no. To generalize from a single decider needs prior reference class ("toin cosses"), getting us into Bayesian subjective interpretations. Frequentists don't have any concept of "probability of hypothesis" at all, only "probability of data given hypothesis" and the only way to connect them is using priors. "Frequency among possible worlds" is also a Bayesian thing that weirds frequentists out.

Anyway, if Omega has amazing prediction powers, and P() can be deterministically known by looking into the box this is far more valuable than mere $1,000,000! Let's say I make my decision by randomly generating some string and checking if it's a valid proof of Riemann hypothesis - if P() is non-zero, I made myself $1,000,000 anyway.

I understand that there's an obvious technical problem if Omega rounds the number to whole dollars, but that's just minor detail.

And actually, it is a lot worse in popular problem formulation of "if your decision relies on randomness, there will be no million" that tries to work around coin tossing. In such case a person randomly trying to prove false statement gets a million (as no proof could work, so his decision was reliable), and a person randomly trying to prove true statement gets $0 (as there's non-zero chance of him randomly generating correct proof).

Another fun idea would be measuring both position and velocity of an electron - tossing a coin to decide either way, measuring one and getting the other from Omega.

Possibilities are just endless.