Hi Katie,
I understand about the furry family members.
I would suggest staying at a place in Jersey City...it's a rocks throw away from NYC.
This website is pretty good:
Official Pet Hotels
I did a quick search and it pulled this up:
Candlewood Suites Jersey City Exchange Place from $99 a night, which beats NYC by a couple hundred dollars a night.
If you scroll down the page in the link above it also lists hotels close to Jersey City (including Newark, Secaucus, etc NJ).
Good luck...