Algorithmically Determining the Rainy Season
This is an interesting statistics problem.  I would do this for starters.  Take all the years of data and calculate average rainfall amounts for each day of the year.  If you're looking for the rainy season, I would use maybe a 30-day moving average on this, but make sure that the average can wrap around Jan 1.  Once you have these values -- 365 values for each site -- I suggest trying a few things:

1) Try out the standard deviation approach.  It's easy to calculate, so you should be able to play around with the particular number "x" in "x standard deviations above the mean".  If it works for you, there's no reason to make it harder.

2) Using "x standard deviations above the mean" really just means that you're looking for a certain value of the cumulative distribution function (CDF) of rainfall.  Instead of using standard deviation, you could calculate these CDFs for each site from those 365 values, and pick the 75th percentile or something.  I'm still thinking through whether this would guarantee that you get a "season"; it's possible that the top-percentile values of the year are nowhere near each other.

3) Plot out a few distributions for key sites: dry midlatitudes, wet midlatitudes, polar, tropics (monsoon-dominated).  If these PDFs look pretty similar, you might be able to figure out a sensible approach.

I think the key difficulty here isn't going to be that different places have different total rainfall, but that different places can have rainy seasons of different lengths.  If the season was always 30 days, I think this approach would be fine.  I'm really interested in seeing those CDFs just to see how different they are.

Let us know what you come up with!