What is the best estimate of a signal given multiple noisy realisations
The term you're looking for is efficient estimator. That's the least-variance (== least RMS) estimator for a given amount of observation.
Whether or not that's the average depends on your noise distribution.
For example, consider the two noises:

continuous-valued uniformly distributed noise
Noise that only takes values $\in \{-\sigma, 0, \sigma\}$ with equal probability.

Both can have the same variance, both are zero mean. Both can be white. But while the average sounds like a good idea for the first one, for the second one you can get a 0 RMS estimator as soon as you've observed three different values by picking the middle one.