With Apple's 1.1.3 iPhone update and the iPod Touch software upgrade, it is now possible to put an icon on your device's home screen that maps to an URL in the Safari browser (called a WebClip).
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.