Normally, the device will generate an icon based on the look of the web page, but there is an option to specify a specific graphic. An icon update was just pushed to TinyBuddy IM, so now you should see this icon on your home screen (if you click on the + then "Add to Home Screen" in Safari):
It is not the best icon, but it is what I could do given my very poor graphics skills. And it should be better than a screen capture of the web page.
(Edit Feb 25, 2008: The icon above is actually a new one done by Justin Kirk, and is light years better than my goofy original icon)
If you previously added TinyBuddy IM as a WebClip on your home screen, you might try removing it, reloading TinyBuddy IM in Safari, and adding it to the home screen again.
