This really depends how close you want to stay to Seattle. Do you want to be in the mountains? Or are you more interested in something coastal?
You could hike the beach on the Olympic Peninsula. A long drive for an overnighter from Seattle. Closer yet are a couple of excellent wildernesses including the Alpine Lakes Wilderness, which is east of Seattle and borders I-90. Many lakes you may camp on for an easy in-and-out.
That time frame of year the weather can be all over the map, so that is a bit of a consideration, especially if you would like to stay on the west side of the Cascades or in the Olympics.
Deception Pass - North of Seattle by a couple of hours. Plenty of campgrounds, you can hike around a lot if you want, a little if you want. Not really a traditional "backpacking" place, it's a nice way to break into backpacking if someone is a bit nervous. You can camp there and do many nice dayhikes.
Spray Park - This is recommended if the weather looks good.(And by good I mean no clouds in the area of Mt. Rainier). A classic hike at Mt. Rainier, definitely in a tougher (but not impossible) class. I'd start at Mowich Lake and head up to Spray Park, and return via the same trail rather than head down toward the Carbon Glacier (don't worry, it's a terrific section of trial, going back the same way isn't a disappointment). Don't camp at Mowich Lake - it's a parking lot. Camp at Eagle Roost (permit required).
If it is colder, then I'd go to Packwood Lake near Mt. Rainier. It has the advantage of being at a much lower elevation (3,200 instead of 6,000 ft or so) but it is a bit of a drive from the Seattle area.
Spider Meadow/Spider Gap ....if you are willing to drive, a very nice (and relatively easy) hike is Spider Meadow near Lake Wenatchee. You gain about 500 feet until you reach the valley...and it is quite spectacular. There you can camp and climb up toward the Spider Gap, a stiff climb if you want. A photo and description is here:
The Washington Trails Association has an excellent site that includes a very good search, including for children-friendly hikes:
Also, they Northwest Hikers message board would be a great place to ask, they would have terrific suggestions as well: