ICS: Use site's bookmark icon
The icon that is displayed on the action bar is called a favicon. It's a low resolution (usually 16x16) icon made to be displayed next to the name in a bookmark bar.
Android understandably doesn't use this as it would look hideous on your homescreen. For a website's icon to show up they need to add another icon: the apple-touch-icon. This one was meant to be used as an icon on the homescreen. Originally for iDevices, but android adapted it. Unfortunately I'm not aware of a way to add an icon yourself if the site doesn't supply it.