Either card will work quite well at 1080p resolution. More than 1GB of RAM is only really useful right now if you are actually going to play the game across multiple screens (eyefinity/nvidia surround). If you are only running 2D applications like your web browser on the second display, you don't need more than 1GB. If you are looking to be a little bit more futureproof, you might want to get the 2GB version, but right now under virtually all cicrumstances the GPU on either card will run out of steam before you get to the point where you need more than 1GB.
I don't have any experience with HP monitors, so I can't help you there.