First off, Chelsea, which is exploding with hotels right now, could be a good choice, as it is relatively near NYU campus. Why not try the Indigo Chelsea? This hits the target of basic accommodations in an affordable neighborhood, while also adding a touch of that Indigo class, so you're not just dealing with fake plants and cheesy lobby music.
As for promo codes, don't waste your time clicking around looking for special offers. For this particular situation, you will be far better off simply calling and negotiating a price for your seven-night stay. Prices, as always, tend to oscillate, and as we head into the fall peak season, it is more important than ever to get a human on the phone and explain the details of your stay.
Now, a few more closer-to-campus suggestions: Bowery House (not to be confused with the Bowery Hotel!) is a pretty unusual, very New York-y—but cheap!—option. Having only been open for a month, the cabin-style hotel is located on the Bowery, a ten-minute walk from Washington Square.
Alternatively, check out the new Hampton Inn Soho, where rooms consistently start below $200 (a deal compared to most other Soho hotels). No, you won't get all the amenities of the nearby Trump Soho (though if you've survived raising a teenager, you probably deserve them), but the hotel's location is un-beatable—walk just seven blocks uptown to reach NYU's campus.