TBH I'd just start by looking at hotels on the online travel agents sites to get a feel of prices and availabilty and how that fits with your budget range. And remember that some sites don't include room tax and other fees until later in the booking process (thought e.g. expedia does right from the start)
December can be expensive as lots of people want to do what you want to do and see it christmassed up and even a lower end hotel can have eye watering rates
I wouldn't discount the financial district. It can be quiet - especially on a weekend - but its close to Statue of Liberty, 9/11 memorial and other attractions and it so can be cheaper over the weekend. I'd also consider Brooklyn hotels near subway stops as well.