The icon that appears next to a URL is call a favicon
This is how you can add the favico to your Google Site
From the
More Actions --> Manage Site -->Attachments
Upload
Click Browse and select your favicon.ico file
Your favicon file must be renamed to"favicon.ico" after it is uploaded. (so if you uploaded some other file name it can be renamed at the google site attachments.
The file format is .ico and has to be a specific size. Go to any of these sites to generate your own one.
FavIcon GeneratorFavicon CCDynamic Drive Favicon ToolChami Favicon ServiceFavicon Generator ORGFavicon UKDegraeve FaviconAnti-FaviconGen FaviconFavigenNot all browsers display the favicon from the google sites.The image at the top was taken using Safari which does display the favicon for me.