I live in Bangkok and the weather is crap here, too! My kids are complaining of the 'freezing cold' haha!!
I second what others have said about the elephant sanctuary on Samui. It is a small ethical one, no riding but you can get right up close and hear individual stories of the elephants. And, one of the most random things ever, there is a video explaining some stuff and it's narrated by Lou Carpenter from Neighbours 😁
Fisherman's Village/Bophut is probably the best area for you to go for some holiday vibes. The Secret Garden has decent food on the beach and they'll let you use the pool there if you want to occupy the children for a bit. The Jungle Club up in the hills has good views and food and is worth a little ride out.
If the weather is really rainy, the only thing I can think of is the mall, sorry!
Oh I forgot one thing - ask at your hotel or Google and find recommendations for how to go to the island with pigs on, it's great! Sorry I can't recall the name but this is my kids' absolute favourite place. You can get a little boat to take you, stop for snorkelling on the way, then hang out on the beach with some pigs, have some lunch, pose for photos on swings etc, then boat ride back. It won't be nearly as Intagrammable in bad weather but would still be fun!
Also the Big Buddha is worth a visit and a walk around the temple grounds. It's not a Thai holiday if you don't visit at least one temple 😁
Sorry your holiday isn't going as planned and the weather is ruining things. It's definitely better here today than yesterday so fingers crossed it improves for you, too!