Hi Barb.
What model cybershot? I had the P100, which could do up to 30s exposures. If yours can do that, then it may be able to capture a streak as the meteor flies through.
Don't use digital zoom (ever), for a meteor shower don't use optical zoom either. You want to capture as much of the sky as possible.
Set ISO to the highest possible (400 on that camera I believe) so it can capture more light, set the aperture to as wide as possible (lowest number). Don't use the video mode - just set it up for a long exposure.
Honestly, you'll have to be lucky though. You'll need to capture 30s after 30s, hoping that a metor flies through at the time you're exposure is going.
It just depends how active the shower is. It might be better to just lie back and enjoy the show, naked eye style!