Express will show a pet icon if it's pet friendly like this example
Name your own price shows no indication if it's pet friendly or not. If you need a place for your pet, I would not do Name Your Own Price, as you could end up with a place that does not allow pets or doesn't allow your pet if they have weight restrictions for example (no dogs over 50 lbs) or you could end up paying a fortune. Even with Express, you won't know the fee up front.
If however you know with certainty which hotel(s) you will get from NYOP, and know they are all pet friendly that's a different story, but that's not going to happen all that much.