I used to live in the Seattle area and it's wonderful! As far as a budget hotel, I've had good results with any Choice Hotels Brand. There's a Quality Inn and a Comfort Suites in the Downtown area according to the website for Choice Hotels.
If you're in the area a day ahead of time, I would recommend trying to go to the top of the Space Needle in Seattle Center. It's about $20 per person I think, but well worth it once you get to the top. Another thing you might enjoy is the Pike Place Market fish market. I don't want to spoil it, but just go to the fish market at Pike Place Market, and that is an experience in itself.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)