I treated myself to a DSC-HX1 a few months ago, when my old DSC-F505V packed up after many years. I got a deal from an online camera store which included numerous accessories, including a 'compatible' (i.e. non-Sony) case.
I've tolerated that case for a while, but it was so ill-fitting that the HX1 moved around inside - not a lot, but in my view, enough to potentially scuff or damage the camera.
So I decided to buy one of these official Sony cases, and I'm very glad I did. It's an absolutely perfect fit for the HX1 - snug, with no room for the camera to move about - but without being so tight that you have trouble inserting or removing the camera.
It comes with a belt loop and a separate, easily detachable external pouch for carrying an extra battery, and a loop to attach it to your camera's existing shoulder strap. No straps are supplied with the case.
The lid is held down by a hidden magnetic 'clip', and the case is padded on the bottom, as well as the front and rear to protect the screen and lens areas.
My only minor gripe is that I would have liked for there to be a little more padding, around the whole of the camera. That said, as it stands, damage is unlikely to occur unless you're very rough - i.e. if you were to drop the camera in the case, it's still likely to sustain some damage.
When my HX1 is not in use for prolonged periods, I'll probably put it in this case, inside that larger, cheaper case the camera store supplied me with, just to be sure!