> What are the independent and identically distributed random variables that are being averaged in this case?

The random variable is simply whether you answered question N on the test correctly or not. For every question there is a probability Pn that any given test taker answers it correctly, and that probability is relatively uniform per-question. (so scores on any test, where the same test is administered to a large numbers of test takers, will be normally distributed. Yes, even if the questions are nonsensical and the answers random)

So an IQ score simply indicates the probability, relative to the general population, whether a test taker will answer simple abstract mathematical questions correctly (where correctly should be understood to have cultural biases).