I have stayed at Doubletree (which is a Hilton brand) and Crowne Plaza in Times Square. Just had a look at the website for DoubleTree and they are doing rooms for $350, which is about �170 - within budget.
Don't stay in New Jersey - pointless when you're only in New York for a few days. Make the most of it and stay very central. I would recommend the Milford Plaza, just one block from Times Square http://www.milfordplaza.com/
I've stayed a number of times at The Roosevelt its excellent lovely rooms and comfortable.
Its on corner of madison and fifth avenue, just minutes from Times Square, and Grand central station. Buses from JFK airport stop at grand central and will drop you off at the hotel, there are some lovely deli's near one almost next door very cheap meals and vast choice. I would recommend it, have been NYC seven times and hope to go again so much to see a great place, Enjoy your trip.
I would recommend the Affinia Manhattan on 7th Avenue. It is right opposite Madison Square Garden, next to Penn St Station and a 2 minute walk from Maceys. All the rooms have a kitchen and as there is a supermarket next door it is really handy. Loads of bars, restaurants all round the hotel and the subway is a l minute walk away but within walking distance of Times Square, Greenwich Village etc. A fabulous hotel