Have you ever looked at the input signal with a 'scope?

That can tell you what the nature of the noise is. If, for example, it has a major component synchronized with line voltage, then sampling at a harmonic of the line frequency will average out that component. If there's some other annoying content, e.g. switching noise from a switching power supply, you can choose your sample rate accordingly, provided, of course, that the average contribution from the noise source is, on average, zero. Since the switcher-generated noise, or the AC line noise, is typically synchronous, you can even go further down the logical sequence and choose a sample rate that allows you to make some assumptions about that noise and correct for it, if you're that daring.

Those things tend to keep on going no matter what you do. It means that you may have to generate some initialization code that allows you to sniff-out and synchronize-with the incoming noise in order to discern its phase and magnitude. Empirical observation of that signal may make it possible to do a good deal of filtering in firmware.

RE