after staying in new jersey last trip, i would strongly recommend considering not staying in manhattan but rather staying in NJ and commuting across. they will save a packet on hotel costs (i am like over the duration of their trip around A$2,000!! and I am not kidding you). don't expect to find anything cheap in NYC over 31 dec holiday period.
 
US$188/night is a bargain for NYC.
 
my suggestion:
 
go to hotwire.com (hotwire 3* are going for about US$180/night in manhattan vs. US$50-$75 a night in Seacaucus meadlowlands), and look for a property in seacaucus/meadowlands. you can catch the 320 bus from 42nd st station to there and it costs US$2.85 each way. 42nd st station is in the heart of times square and is only a 15 min bus ride away. buses go there regularly from 5am-1am each day. Trust me, it seems like ages away but the US$130/night they'll save each night can more than make up for a few drunken taxi rides back after some great nights or the bus rides - seriously they are 15 min bus rides and they leave 42nd st bus station (whcih is part of the subway station every 20 mins on the dot, from 5am-1am daily). 
 
For hotwire properties and matching the codes, try the key word "jersey city" or "new york" respectively in hotwire. To find out which properties they are use: http://www.betterbidding.com
 
If they are desperate in staying in manhattan, try:
 
http://www.hotelqt.com  (for any night they can get, very very good value and a great property, use tripadvisor.com to get reviews)
qt is in the middle of times square and includes comp breakfasts.
other than that, i can only suggest, looking for reasonably priced properties at:
 
 
WIth easyclick travel the bargains lie in the "too good to name", where once you're in the section where they explain the property, copy the dialogue, paste it in google, and more often than not (actually 98% of the time) it will spit out the same dialogue on the actual hotels site and you'll know which property it is that they are saying is "too good to name".