I'm going to cheat a bit and just mention it: Pandemic rising tide:
It's a board game in which the Netherlands is slowly being flooded and it's up to the players to build dikes, dams, pumps and what have you. While it might not explicitly be raining, I kind of imagine it does. Heavily.
As for video games...I'm constantly thinking "heavy rain! heavy rain!" but I haven't played it. And I've got no idea if there's rain in it to begin with.
So two others, depending on how you slice them. One is probably a strange one: Sleeping dogs. Yes, it has rain in it. But due to the engine being used, it looks as if all the players are constantly wet (and oily) all the time. I mean, look at this guy:
He looks so wet rain bounces back from him.
As for the other one: Ori and the blind forest. Sure, it hardly rains outside the introduction, but man...you can say that it pours in that introduction!
