You want to avoid the ulta thin cameras as they are difficult to hold steady.
A larger camera or a small camera with a bit of weight to it will help you hold it steady.
Also, you should use the viewfinder instead of the LCD, most of the time. With the viewfinder, you can hold the camera against your face when taking a picture. Your face (head) is steadier than holding your arms out in front of you to see the LCD screen.
When possible, lean your shoulder against a post, tree or wall.
Take a look at the Sony P93. A 5 megapixel, 3X zoom with a good movie mode. According to the specifications, if you buy additional Memory Sticks for this camera, be sure to buy the Memory Stick Pro. The Pro stick is faster and is required when using the best quality movie mode.
Here is a link to a review:
Does someone have a suggestion for a digital camera for someone whose hands shake slightly? I would like it to take mini videos and be at least a 5 pixels.