Ideas: Not great ideas, but this is what I would try.
Remove any memory cards and try starting it. (Try a different memory card.)
Try it with new batteries or plugged into a power source. (Are the batteries the right ones. Some cameras are finicky.)
Make sure nothing is stuck in the plug that connects it to a pc or dock.
Try plugging then unplugging it into the pc cable or dock a few times then restart it.
Play with any knobs or switches. Cut them off and on try starting it with settings in different positions.