I heard if it rains on the second (or first?) day of Shavuos, it will rain at some point during each of the next 40 (?) days.
It definitely has rained a decent amount, but is this a real thing? and if so what is the source for it?
I haven't heard the "some extent" part. People have told me that this is an absolute - 40 days of non-stop rain. It's obviously false. While Houston got the equivalent of their 40 days of rain (and more) last year, I know of nowhere in the world that has had 40 days of continuous rain since the days of Noach. It's possible that the expression means, more simply, that there will be rain at some point of each of the next 40 days. That is possible and does happen in numerous tropical climates. I gather that during the summer it rains at some point daily in Florida during the summer, esp. But, I think if you asked a meteorologist, I don't think she would correlate this with Shavuot.
As for the source of this idea, I have heard these:
Torah is compared to water. Moshe was on the mountain for 40 days, beginning from Shavuot, receiving and learning the Torah. So, it must have rained up there for 40 days.
The giving of the Torah was accompanied by a "thunderstorm" on Shavuot. The Torah never mentions anything about it stopping. So, people assumed that this storm lasted the entire time that Moshe was on the mountain.
Take the above as mere hearsay. This may be the actual source or not. It's just what I have heard in the past from a few people. I have no idea where they heard it from. It's often difficult to verify the source of superstitions.