It's been 10+ years since I was on these hikes but as best as I remember:
As mentioned above, you can hike up Mt. Si in any weather. It's just a trail hike thru forest - no views until you get to the top. It is very popular on weekends. Gains about 3K in 4 miles. Take I-90 east to North Bend. If there is snow on the trail it may be packed into ice so crampons can be useful. https://www.google.com/#q=mt.+si
I think Si and Tiger are on state land.
Closer to town, at exit 20 on I-90 (just past Issaquah) is Tiger Mountain. Gain is about 2K in 3 miles. There are several trails at the start so look at a map or ask which way to the top of Tiger Mountain. http://so-seattle.com/article/hiking-summits-tiger-mountain
For more of an alpine hike, there is Granite Mountain at exit 47 on I-90 east of Seattle. I think this one is on Forest Service land so Obama's shutout could affect access - but it's only 1/2 mile from the freeway exit to the trailhead so could walk it. Gain is 3.8K in 4 miles +/-. Starts in forest, then eventually crosses a gulley and breaks out into thinner trees and brush - if snow is deep DO NOT cross the gulley due to avalanche danger. Get off the trail and go up in the trees keeping left of the gulley. Soon the trees thin out - if snow is deep and soft go back due to avalanche danger. Top 1500'+/- is above tree line. Good views from the top except for freeway below. http://www.wta.org/go-hiking/hikes/granite-mountain-1
NEVER leave any valuables in your car at a trailhead.