Check out this link: www.digitalcamera-hq.com/digital-cameras/fuji-finepix-z1_question.html?qid=4289
After the basics (cleaning the lens, replacing the batteries, etc.), try dropping the camera on a carpeted area from about 2-3 feet. It worked for my Fuji A500 and 90% of the responders on the link.
Everything on the web said it would cost ~$100 to fix.
I was in the process of researching new cameras to buy and had nothing to lose by dropping it on the floor.
The error may be caused by debris in the zoom mechanism preventing it from working.
Dropping the camera loosens the debris and voila!