I'm 27 week pregnant with a 19 month toddler and live in Repulse Bay/South Bay, so I can relate to you on being in a need of finding things to do in the area!
I often take my boy to Repulse Bay playground/beach early morning or late afternoon (it is too hot to go there in the middle of the day). If we want to swim, we usually head to South Bay Beach (water is clearer there than Repulse Bay). He also enjoys the Repulse Bay shopping arcade, going up and down on the lifts and escalators, watching the turtles and fish in the pond, having fruit juices and sandwiches from Pacific Coffee, and running around inside Marketplace. If it rains we often go to Stanley indoor playroom as others have mentioned.
There are not many activities to do I can think of in this area, so I take him to gym classes in Central. I heard you can become a week-day Repulse Bay Club member at 109 Repulse Bay Road if you want to use the facilities and take kiddy classes.