I can't recommend a particular hotel but I can recommend an excellent website. It's this one:
https://www.hostelworld.com/
(NB: Don't be put off by the word 'hostel' in that URL. There are plenty of budget hotels listed on there too. Simply conduct an initial search and then use the filters to find them).
There are some real bargains to be found on that site. For example, three years ago I stayed in a wonderful little hotel in central Madrid (ideally located and superbly clean, with air con, free wi-fi, TV, etc) for €22 per night.
Once you've found somewhere that looks interesting, check the location carefully (because, for example, some of the 'London' hotels listed on the site are nowhere near the city centre), read the reviews on that website and then look for more on TripAdvisor. If you're prepared to put a bit of time into your research you should be able to find a really good, but really cheap, hotel in NYC.