Whole apartment rentals (under 30 days) are technically illegal in New York City (and now Jersey City as well), so the only legit options are a room in a shared apartment where the tenant is present. Probably not what you want.
There really aren't any areas to avoid from a safety area in Manhattan (where hotels would be), so it's really just a matter of convenience. So anything from Midtown East/West down through Garment District/Chelsea/Flatiron/Grammercy would be good. Greenwich Village/Soho/Tribeca is fine too, probably a bit more "hip" (for better or worse) but all a little further downtown. I probably wouldn't bother with the Financial District, it's kind of dull.
I would caution that December is prime holiday tourism season and the city is very busy and prices are likely to be higher than normal. I'd probably avoid any lodging directly in the vicinity of Rockefeller Plaza as the streets will be packed with people going to see the Xmas tree.