In Mega Man X2, it rains in Wire Sponge's stage. The stage is called Weather Control. There's no rain sound effect, but there's a wind mechanic (everyone's least favorite game mechanic in a platformer).
In Pocky and Rocky, it starts raining right in the first level but clears up not too long after it starts. If you haven't heard of this game, look it up! It's on the SNES and there was a sequel as well. You control your character with a top-down view like Zelda, but it plays like a shmup. I haven't played much of it, but it has some beautiful 16-bit graphics. The dive move that you can do is a lot of fun.
Another example I can think of is Shenmue. I don't think the rain comes at a specific event but there are definitely days where it's raining while you are exploring, and you will see NPC characters out and about with umbrellas in hand.
If I recall correct, it will occasionally rain in Grand Theft Auto III. The game has a day/night feature as well as other weather effects. And, of course, every game in the series after GTA3 has those features as well.