Suppose you have a discrete random variable $X_1$ with known probability mass function. I guess that choosing a variable drawn from the same pmf would be the best way to guess $X_1$ assuming all experiments are independent. If this is true, what is the name of this theorem?
By the most usual criteria of "best guess", that's not an optimal strategy. For example:
If you want to miminize average square error : pick E(X)
If you want to miminize average absolute error : pick median(X)
If you want to miminize probability of error: pick mode(X) (maximum value of pmf)
To use random strategies for guessing can be optimal is some scenarios where there is an opponent who can vary his strategy observing yours (eg. rock-paper-scissors), but that doesn't seem your scenario.