They're not unique to Seattle, but I've never been anywhere with beets on the menu so much! Seattle also seems really nuts about goat cheese. Both are locally produced.
I love regional hot dogs, so I'll agree you should check out a Seattle-style dog (cream cheese and sriracha).
Geoduck and Washington Rieslings (not necessarily together).
As mentioned by other posters, there aren't a ton of really uniquely Seattle foods like in other places, that I know of. I'd suggest Dungeness crab, some oysters, and geoduck. If you haven't had sweet smoked salmon (salmon candy, Indian candy, etc.) it's not unique to Seattle, but it is pretty tasty. Check out the Pike Market for some excellent smoked salmon. Chowder was another good suggestion - try Pike Place Chowder's offerings...some of them are outstanding. Other than that, maybe some Washington wines - maybe try some of the Rieslings which Washington does pretty well but are often undervalued. http://www.nytimes.com/2009/09/30/din...
You might also check out Fran's chocolates - lots of people love them.
I like all the suggestions, but I don't think they quite get to the OP's question. None of these are iconic-ly Seattle like Chicago Dogs, Philly Cheesesteaks, Beek on Weck, Garbage Plates, Hangtown Frys, Lobster Rolls, etc are for those respective regions.
The Pacific Chowder (one with bits of salmon and crab in it) is close, and maybe smoked salmon (the lox style kind of, but definitely the dryer hard smoked style) is a pretty solid one.
Oddly enough, I might nominate one of the late night hotdog carts with Cream Cheese Hot Dogs for this. I've never seen that anywhere else and it's pretty common for the Fremont and Pioneer Square hot dog vendors.
*Salumi Salami- only open t-f
*Regional Berries and Fruit (Pike Place Market or seasonal dessert menus)
*Vietnames (Green Leaf or Tamarind Tree) in the international district
*Something fish related- Like cheap fish and chips on the Pier at Ivar's, Pike Place Chowder- Clam or Scallop Chowder, or Fish (salmon?) at a place like Steelhead Diner- sources from Pike Place Market- or Market Grill
*Gastropub food- Quinn's, Spur, or another
*Local coffee- Cafe Vita (lower queen anne or capitol hill) or another (Victrola, Stumptown-in Portland too, Vivace, etc)
A local food cart (Skillet, Marination Mobile, or Maximus/Minimus)- they are gaining momentum in the city and are a new phenomum in the last couple years- their websites say were they are located each day
ALSO- Eat your way thru the market one morning and afternoon- be aware most things don't open until 10am most days- All worth trying= daily dozen donuts in a paper bag, crumpet shop, svlenda bakery, piroshky piroshky, market grill (fresh fish grilled sandwiches) , three girls bakery, beecher's, bavarian meats, delaurenti's (store/deli/gourmet), etc
Any complete Seattle trip deserves some serious time wandering the Pike Place Market. It's not the same old Rouse Mall Food Court thing you might have learned to expect. For one thing, chains are not invited, so you will find no Quizno's or Cinnabon here. You will tell grandchildren that you had Cioppino at Jack's Fish Spot, a freshly grilled Andouille sausage at Uli's Famous Sausages (how did a German dude ever cotton to this?), Carne Asada tacos at El Puerco Lloron, a snappy little salad at DeLaurenti's, and the list goes on - a wonderful way to spend a day.
I have to agree with the oyster/crab recommendation. Elliott's Oyster House, on the waterfront, has an awesome oyster happy hour every day. I think its from 3-6p. Oysters start off super cheap, like 25/50cents?, and go up a quarter every half hour.
If you don't have much Vietnamese food where you live, Seattle is packed with Pho joints. It is a tasty meal on a cold, Seattle day. I like Pho Cyclo in Capitol Hill, but there are a million places to pick from.
Agree about the oysters and crab. And salami or other cured meat from Salumi near Pioneer Square. Given the time of year, also consider the humble Washington apple from a local vendor at Pike Place or one of the other farmer's markets. I've been munching my way through a fresh 5-lb bag of Honeycrisps from Anderson Acres in Yakima all week. The apples from Eastern WA will cause you to re-think the fruit entirely.
I'm not crazy about most of the Eastern Washington apples -- they tend to be monocrops of the sort of generic varieties you find in supermarkets or exported. My favorite apples are the varied, misshapen heirlooms you're more likely to find at small farmers' markets. Jones Creek up on highway 20 (where I just picked 50 pounds of all sorts of apples), places like that.
Other recs: Northwest style smoked salmon (Pike Place market or Wilson Fish at the Ballard farmers' market), fall's wild mushrooms, Ethiopian food.
This kind of stuff depresses me:
Eating dried-out, cleaned-out, hacked-up restaurant crab that's being tossed around by the masses in a restaurant chain, and then being charged an arm and a leg for it (pun intended).
Go to Ranch99, get a dungeness out of the live tank for $3.99, and have them steam it for you for free. Then it's really fresh, really tasty, and you get to enjoy the tomalley and the crab butter.
Anthony's and Arnie's are great if you're over 70 and can't taste your food, and have enough money to throw away on it.