Cattle Point Lighthouse
This is on the grassy side of the island and you can hike over to the lighthouse pretty easily if you have an energetic kid. If he needs to be carried though I'd skip the lighthouse part and just go play at the beach in this area. There are bunnies in this part of the island too - especially in the evenings.
San Juan Island National Park
One of our nation's newest National Parks!! This was a State Park for a long time and it made the upgrade list a year ago or so to National Park and got itself a little face lift in the process. You should go check it out. It's on the west side of the island where you're most likely to see whales. http://www.nps.gov/sajh/index.htm
The Whale Museum
Speaking of whales. Go check this out - it's pretty cool and you can walk there from downtown. There's neat gross stuff like a giant whale brain embalmed in a jar and other things your kid will think are the coolest things ever.
American Camp/British Camp/Pig War
There's some history to know about the island. It was disputed territory between the US and The Queen back in the day. There was a pig. Lives were lost. Things were stolen. Dozens of hours were spent fighting. Dozens. And the military camps are still on the island. It's kind of cool to read up on the history and then go see the camps. They're also State Parks. Go. Learn. Feed ducks at the park. Wear a coat.
Fourth of July Beach
This is out by the lighthouse on South Beach and if you like sea glass it's the best beach to find sea glass on the island. At one time there were rum runners in Canada during prohibition and when they would get caught they'd dump the goods overboard and the glass washed up on Fourth of July beach. As the name suggests, this is also a good place watch fireworks on the Fourth of July.
Crow's Nest Coffee :) Mary and her husband Jon own this place and one of them is always always there. They brew Vivaci from Seattle and it's easily the best on the rock. Their breakfast sandwiches and burritos are AMAZE BALLS.
http://crowsnestcoffee.com/ - ok so they need a dev team. We love them anyway.
RESTAURANTS!!! (You can also ask Mary & Jon for recommendations)
The Cask & Schooner is damn good and kid friendly and right there in town. If it looks too good to be true don't be frightened off. It's perfect. The food is delish and trendy and the wine list is killer and the prices won't kill anyone. Do that if you're tired and all you can do is stumble out of your hotel and beg to be fed.
If you want a fancy date night dinner you can always go out to Roche Harbor and you'll get a great meal and a beautiful view.
The Back Door Kitchen is heavily recommended as some of the best food on the Island. I don't know if they're open in winter. Check into that.
Other places worth the wait and the price:
Duck Soup Inn - middle of the island may not be open in winter
The Place - down on the ferry landing
The Market Chef - also highly recommended
Chill (cheaper) food:
Mi Casita Mexican Food
The Rumor mill
Herbs Tavern - classic old timer salty dog joint
The Doctor's Office - Ice Cream Shop :)
OK - I can keep going but I should NOT do that. Post in the comments if you have questions.