I went in December when it rained for a couple days. Obviously, doing any indoor and semi-indoor rides while the park is less busy is lots of fun. Just make sure your shoes have good traction, and/or walk super-carefully.
It sounds ridiculous, but if it's COLD and raining, Innoventions is heavenly. The CM letting us in emphasized how warm and dry it was. I'm sure none of them were used to having half as many people as they had that day and they all seemed really excited.
A whole group of people were in the living room of the "House of Tomorrow" as if it were their own, watching WALL-E. I found it hilarious.
They even had some Winnie the Pooh characters up on the top level so you could get your pictures with them.
Space Mountain wasn't a walk-on (and there was some water dripping or something so they did end up shutting it down after a while), but the lines were definitely very manageable. (Super-slippery throughout the queue, though!

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DCA was pretty much literally a ghost town on a drizzly night. There were at most maybe 10 people ahead of us in line for TSMM (meaning that it wasn't technically a walk-on but it was darned close.)
If you like to take photos, you can get some lovely shots in the rain if you like.
But basically, almost anything you like to do that is covered or indoors is just great in the rain. If you are dressed for the rain you can technically do a lot of other things too, but that depends on your tolerance for getting wet (and if you do an outdoor coaster, your tolerance for getting pelted in the face by raindrops at 30 miles per hour.)
The only thing I found that wasn't any fun at all in the rain was trying to get something to eat at the Hungry Bear Restaurant around lunch. It was packed and the lines were very long and pretty wet, even with an umbrella. To tell you how long the lines were, one woman was down in the covered area saving two tables for her party (meaning it took us a long time and some trouble to find a place to sit). By the time her party came down with the food, we were literally on our last bite of lunch.
Have a great time!