This is a problem, isn't it.

I wouldn't head up to the Peak during the day; you won't see much with the clouds and fog. But if the cloud cover lifts just a bit in the evening, then with the lights below the Peak is still much worth a visit then. But bring an umbrella.

There are the obvious indoor places: malls are everywhere. The HK Museum of Art right on the TST waterfront makes a good rainy refuge. As of course would very long and pleasant meals.

Consider too a ferry trip to one of the Outlying Islands. It may be too rainy to walk around and you'll have to head right back, but the trip through the harbour is always great no matter the weather. I'd skip the Bun Festival myself, but I don't particularly like rain, or bun festivals.

Check the HK Observatory website for the current conditions, and understand the warning signals.