Very open-ended but I’m going to be in New York for a weekend and have a lot of time to fill-- what should I see? I’m interested in seeing things that are not so obvious (like, for the most part I can look up museums myself). Even if it’s just something neat to look at from the road, I’m game. Or just a neighborhood that tends to have a high density of cool stuff-- I’m definitely into soaking up the vibe of a given area. I know this topic is vague to the point of unhelpfulness, but feel free to project your own interests onto it. Extra points for odd, unique things. All five boroughs are good to go.
Ride the Staten Island Ferry, it’s free you know 
This is the best thing a tourist can do.
Also go to the top of the Empire State Building after dark. It’s breathtaking.
I think they reopened the Statue Of Liberty. If you can climb to the top, do it. It’s not much of a view at all, but the fun is in the climbing.
It’s technically a museum, but The Cloisters are quite cool. Plus, from there you can walk over the George Washington Bridge to New Jersey.
From near Fort Tryon Park, you can take a bus (used to be the #12, but I’ve been out of NY for 12 years so confirm this) over to City Island, which is like a little town on the Maine shore, except it’s still part of the Bronx. Cool place for seafood!
Near the World Financial Center, and two blocks West of the WTC, there is an Irish Hunger Memorial that is really cool.
You can walk the labyrinth in Battery Park, though I can’t personally vouch for it.
I really want to see the new High Line Park, and I hope I can get there soon. It is not far from my favorite Tea shop in NYC, Tea and Sympathy. In fact, the whole West Village is a fun place to walk around. Various Bookshops, Kim’s Video, Murray’s Cheese Shop, the historical Stonewall Inn. Speakeasies?
Don’t go to the Strand. It’s way overrated.
Seconding the West Village area. You can just walk the streets, some still cobblestone, and see a bunch of interesting stuff. Look for the historical markers on the old buildings. No shortage of restaurants/cafes/pubs, etc. either.
If you actually venture out to Queens, you should go to the largest spa in NYC, Spa Castle:
http://nyspacastle.com/eng/main/main.php
It’s a Korean spa. There are dozens of saunas, hot tubs, outdoor hottubs, hot-tubs where you can lie down and watch TV, a food court, more saunas, etc.
If you go during the week, it’s only $35 to spend the day.
Another kind of unusual museum you might not otherwise find is the Tenement Museum. I found it pretty interesting.
http://www.forbesgalleries.com/
The Forbes Galleries contains Malcom Forbes considerable collection of toy soldiers and toy boats. Free! Strange!
They used to also have a collection of Faberge eggs that was pretty cool, but that was in the '90s. I don’t know if they still have them.
I’ve never been to NYC other than to change trains or planes, but a place I do want to see when I go there as a tourist is the Morris-Jumel House. In addition to being one of the only 18th century structures left in Manhattan and the oldest house left in the city, it was Washington’s HQ during the Revolution and was later the home of Eliza Jumel, a prostitute in her youth/courtesan in her younger years/society wife later/notorious figure in 19th century NYC who upon becoming a very wealthy widow of around 60 married the only man in NYC more notorious than she was, Aaron Burr. (It didn’t last- she divorced him when he began spending her money.)
There are also guided walking tours of ghosts and unusual attractions that can be googled.
“Strange, interesting sight” + New York City = The Panorama of the City of New York.
Yeah, the west village is definitely fun to browse around. Stop by Aphrodisia which is a ridiculously amazing herb and spice store. If you need some eye of newt, you’ll probably find it there.
Seconding the Tenement Museum and the Forbes Galleries.
And I always say this in these threads–the public library on 42nd St. Go to the main reading room. Look up.
These are all awesome ideas! I’m going there in January and love unusual but awesome stuff.