Actually, with the H5 you can change the ISO settings in all modes except for one - the 'Green Box' or full auto mode. I'd encourage you to never use the full auto mode for just that reason.
if you're not that knowledgable about cameras, try the 'P' mode on the dial. It's much better than Auto. It actually works just the same - the camera still has all automatic settings, and you can still just point and press the shutter button if you want. The difference is, this mode will allow you to change the ISO, the White Balance, the focus modes, the metering modes, etc. if you so choose. You can change the ISO in P mode, S mode, A mode, and M mode, and also in most of the 'scene' modes. Just not the green auto mode.
Your H5 will have Auto ISO, as well as 80, 100, 200, 400, 800, and 1000. In auto mode which it sounds like you've been using, the camera will often choose a higher ISO in low light situations rather than a longer shutter, because they know many people who don't know anything about photography may not be using tripods and therefore will end up with a blurry shot.
For the best night shots, you want to go to the lowest ISO, and use a longer shutter speed while putting the camera on a tripod or level surface...and it's usually best to use the self-timer to let the camera snap the photo without your hands on it, so it doesn't shake at all. The H5 can take excellent night shots - I've taken hundreds with mine over the years, including tons in Disney.
If you absolutely MUST get a handheld night shot, such as for a moving subject, then you use higher ISO to get a faster shutter speed. But otherwise, keep it lower and let the shutter go longer for best results.
Oh, and for more info from other users of this camera, check out the Sony Talk Forum board over at dpreview.com. There are lots of H-series users there.