You can still do all those things - just bring an umbrella. If you take the baby out - bring a couple of towels in a light backpack. The first towel will help shield baby from scattered water drops and keep baby somewhat dry. The second towel - always good to have a backup.
If your friends like the noise and want to people watch - go to ANY large shopping mall (golden eagle, nanping jie, etc) and sit in one of the coffee shops/cafes.
There's also the newly opened (?) Flying Tigers museum and the military museum near green lake.
As a previous poster mentioned - just strolling around green lake in the rain is charming and romantic. There's a large upscale tea shop near the entrance to the green lake road loop (it's a one way street, so there's technically only one entrance). They sell nicely packaged small lots of tea, so you can buy several different kinds to sample, enjoy, give away for gifts.