I would stay close to Union Square. I think that's where you will find the most activity. It appears that the Union Square ice skating rink was open on Christmas Day last year. Assuming the same holds true this year, that would indicate that people/tourists will be around. I assume a number of restaurants in the area will be open to cater to tourists, at least.

Expedia is showing decent (4*) hotels right in Union Square on 12/25 within your price range. I bet you could get an even better rate on Priceline.

I would stay away from most other areas, including the Financial District, as they will probably be dead. Staying in Chinatown sounds kind of sketch to me, but if that's your thing...