[steven]

I wish I could offer more help, but I've only been to Osaka a few times and an indoor place that comes to mind to me is the aquarium... unfortunately it's all the way south.

I really liked Arashiyama and would like to go there when there is a light rain. There's a lot of bamboo there and a ton of monkeys. I'd only be afraid of the ground if it got muddy... so it's probably not the best choice. It'd be an adventure though! They have a lot of shops at the bottom of the mountain (and a beautiful bridge overlooking the river). You might get some omiyage shopping done there. I know they have a lot of museums in Kyoto, too. I can't remember there names as it's been years.

If I were you, I'd bring some rain boots & a change of clothes to allow yourself to be a bit more adventurous rather than staying indoors. That's just me though (I think I'm developing a complex where I like to be otudoors more than indoors recently though so take what I say with a grain of salt).