Pannett Park is lovely - museum full of weird and wonderful curios, and best of all for your DS there's a great pirate-themed playpark next door.
If he's into pirates he'll also love the reduced-scale replica of Captain Cook's ship, the Endeavour, you can take a 25 minute trip in it out of the harbour and back. It runs from the quayside near the amusement arcades, I think it's free for kids and only a few quid for adults.
In the same area (outside the RNLI museum/shop) you can usually catch 'Elizabeth the Steam Bus', the last road-going steam bus (in the world?). Takes a 20 minute tour around town, the couple and their son that run it are steam buffs, and the husband drove the 'Hogwarts Express' in the Harry Potter films. Sometimes they add a 'Thomas the Tank Engine' style face to the front of the bus, and the kids like getting their photo taken there.
From the RNLI museum/shop walk up the winding west cliff and kids usually like seeing the big whale bone at the top of the cliff and the statue of Captain Cook. Up there on top of the cliff there's also a crazy golf.
The Pavilion on the west cliff shows films on certain days, there might be a kids film showing as it's the holidays. Other rainy day activities might be story time sessions at the library (it's near back of the train station), and I think there's also a pool at the leisure centre round there.
Cafes - my kids (DSs aged 4 and 8) love going to the cafe down the lane from the Shepherd's Purse shop on Church Street (the street that leads from the 199 steps back into town). Not the chicest place but very welcoming to families (squashy sofas, great choice of kids food, including veggie options), great range of home baking and a huge treasure chest full of toys to play with.
There is a lovely bookshop on Church Street that has a great kids section. They sometimes have author events, and story sessions.
Days out - North Yorks Railway is great. even if you don't go on a train, go and see the train when it's in the station, with its huge puffs of steams and smoke - great fun for little (and not so little...) boys! If you want to go to Scarborough you'll need to get the bus. Nice beaches there, and a Sealife Centre in the north bay. There's a miniature railway that goes from the Sealife Centre into a park.
Hope all this isn't too late for you!