Cheap hotel or hostel to stay for one night?

Definitely check out Hotel 414. It's in Hell's Kitchen which isn't the best neighborhood but it's inexpensive and really cute. It's very clean and the staff is great. There's also an adorable little courtyard where you can hang out and socialize with other guests. 

Also check out The Pod Hotel. This is more of a true hostel but it's clean and very cheap. I've never stayed here but I have a friend who stays here a lot and says it's great. 

Check out The New York Loft Hostel. Great Brooklyn location right next to a subway line. It has shared and private rooms with an big open canopy backyard common area. They also provide breakfast. 

The Jane Hotel is a good option, only 99 dollars for a night's stay

I would check-out airBnB. I've found that you really can't beat their prices when it comes to NY. Try looking in Brooklyn Heights. While it isn't Manhattan, you have have 5+ subways within a 10 minute walk and it is just one stop into Brooklyn. Plus that area is fantastic. There are probably good ones for one night.

They don't exist! You're in New York so cheap doesn't really happen in these parts.

Look up places in Astoria, Queens on airbnb - it's the cheapest area that's close to Manhattan (albeit far from Brooklyn).  If you're committed to Brooklyn, cheaper areas you could find on airbnb for Prospect Heights (possibly).

If you are easy and not afraid to wait last minute, I recommend waiting 5~7pm and then using the Name Your Price feature of Priceline. I generally do that when there and grab 4~5 stars hotel for super reasonable prices!

There are a couple on The Bowery that I know about. I actually stayed at The Bowery House and would recommend it for one or two nights. The Bowery's Whitehouse Hotel was recently written up here: http://ny.curbed.com/archives/2014/03/26/spending_the_night_inside_nycs_worst_hostel_cost_45.php?skip_mobile_check=true.

At the same time I found The Bowery House I found The Jane Hotel. I'd like to stay there next time.