If you're at the Seattle waterfront with your kids, be sure and do "Ye Olde Curiosity Shop". Mostly it's a tourist shop, but it's got some old mummies, and jars of 2 headed pigs and other such curiosities to view. Used to love that place when I was a kid; it's like a Ripley's Believe it or Not shop.
I would not bother with the Underground Seattle tour, especially with kids in tow; my experience is they just aren't that into it.
Ferry rides are always good. If you are doing the Seattle to Bremerton run, there's a Navy submarine museum thingy out near Bremerton/Silverdale. I didn't go, but my brother, who loves all things military, told me it was awesome.
The ferry boat trip out to Orcas Island is nice too, and at the state park there you can take a short hike up to a viewing point from which you can see (on a clear day) up to Canada to the Cascades to Mt Ranier and sometimes whales. However, a trip out to Orcas Island is at least a whole day affair, and the ferry charges are probably quite spendy these days.
There's also the Museum of Flight over near the Seattle Boeing Plant, if you're into airplanes and such.
I've never been to the EMP (Experience Music Project) but my kids have gone and tell me I must do it one day.
I remember going out to the Ballard Locks when I was a kid and that always seemed like something exciting - boats, ships, fish ladders, gardens and Norwegian culture. I think the official name is the Hiriam Chittenden Locks.
The Space Needle is OK, but I would rather go to the Pacific Science Center which is not far from the Needle. There's lots of rides, games, and eateries at the Seattle Center for kids, also a great Children's Museum at the Center. You could easily drop a lot of money at the Seattle Center.
Some of the State Parks are pretty cool. My kids always liked the old Forts, they liked running through the underground bunkers and mazes. Not me, I get claustrophobic. If you do visit some of the old forts with your kids, make sure they have flashlights or glow sticks, and a whistle so they don't get lost in the maze. At many of the state parks, if the weather was decent, I would take my kids there for kite flying and a picnic. That was an inexpensive fun way to spend a day.
Have fun!
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We cannot control the evil tongues of others; but a good life enables us to disregard them.