What kind of lighting are you using? I have a cheap camera too... can't find it at the moment, but when it does turn up I may want to try some night shots. Thus far, I have only tried it without a good source of light and it is dreadful.
I remember back in Photography class being floored when I learned there was a 3200 ISO film that could capture pictures in near absolute darkness. Digital point and shoot just doesn't pull off dark shots.