In this problem, you have a real valued quantity which has an unknown distribution, and you want to estimate the error on the average from averaging N independent draws. This is an application of the central limit theorem.
If you choose $N$ independent picks from a distribution $\rho(x)$ with zero mean and finite second moment, meaning that
$$ \int x^2 \rho(x) = \sigma^2$$
Then add the picks together to get $X$, you get that the sum is distributed as the $N$-fold convolution of $\rho$ with itself. This eventually converges to a Gaussian of second moment $N\sigma^2$. This means that the error in taking an average over $N$ steps is the width of the distribution of $X/N$, which is
$$ {\sqrt{N\sigma^2}\over N} = {\sigma\over \sqrt{N}}$$
In other words, the error is Gaussian distributed with a width that falls off as the square root of the number of trials. Your error is the $\sigma$ of the distribution divided by the square root of the number of trials.
If your trials give you $x_1,x_2,...,x_N$ for x, you calculate the mean of these, and call it x:
$$ x = {\sum_i x_i \over N}$$
Then calculate the average square of $(x_i-x)$ to get a best-estimator for $\sigma$
$$ \sigma^2 = {\sum_i (x_i - x)^2 \over N} $$
Then your error is $\sigma/\sqrt{N}$. This is the best estimate to make in your circumstances.
The one thing you have to check is the convergence to a Gaussian. This means that you make sure that the roughness model gives a distribution of intensities where the $sigma$ is not dominated by rare events, so that the estimate from N trials of $\sigma$ is reliable. In principle, you could have a contrived rough surface model which produces ridiculous things on very rare occasions--- for example, suppose the roughness occasionally conspires over long distances to be a nearly exactly periodic sinusoid. This makes the surface a diffraction grating, with extremely sharp peaks at certain directions. If you are looking at the direction where the diffraction peak occurs, you get a huge outlier in the scattered intensity for a diffraction grating configuration, and this might occur in too few trials, so that the peak might never show up in your limited number of trials.
In practice, you just make sure your roughness model is not conspiratorial (so as to make periodic gratings) and for this it is usually enough (aside from contrived nonlocal roughness conspiracy) to check that the distribution of intensities doesn't have a power-law tail. You cannot do anything rigorous without a description of the roughness model, so that you can show that diffraction grating conspiracies don't happen with any reasonable probability, so that the second-moment is well described by what you see.
This is usually true, absent obvious tail phenomenon (for example, you see 2 trials out of 200 that dominate the variance) so doing a central limit theorem error analysis is good enough.
