I wouldn't worry about staying near Times Square to be honest with you - it's a great place to go and see, but you can stay anywhere in Manhattan and get there very easily.
Few places I've used and would recommend:
The Pod Hotels - great budget hotel. The rooms are small, but comfortable enough for two. You won't be spending that much time in your room anyway!
Herald Square Hotel - I prefer to stay nearer Herald Square (close to Penn Station/Macys/Empire State Building/Koreatown). It looks a bit old-fashioned, but the rooms are nice and the staff are very friendly.
Jazz Hostel - stayed in one of the older hostels, which has now closed down. If you're looking to stay on a budget, I'd recommend them. They do 2 bed dorms if you'd rather not share.
If you're doing a 10 day trip, there's a few things you can do outside of the city. Atlantic City has got a huge selection of outlet shops, plus a nice boardwalk out by the beach (weather depending!)
Niagara Falls is a great trip to do. Fly to Buffalo, then take a taxi to Niagara. There's not a whole lot to do there beyond the falls and the Skylon tower, the town itself is like an old seaside town (think amusement arcades, tacky museums and lots of knickknacks for tourists) - you could knock it out in a day if you left early.
Boston is only a few hours away - in terms of hotel pricing, it's very much on a par with New York, so you'll have to hunt for a bargain. I've not been myself, but friends have recommended it for a 2/3 day stay.