A favicon is the small icon that appears in your browser next to your site name in the tab, bookmarks and browsing history listings.  Learn more at Wikipedia-favicon or Favicon cheat-sheet.

browser tab with favicon

To customize your site with a personal favicon, you create a favicon.ico file and then load it into your Google Site:

First you need to create a favicon.ico file.  Google Sites will work with the basic 16x16 pixel .ico file and also supports higher resolutions, regardless of the size it must be uploaded as a single file named favicon.ico.
  • There are numerous free websites that enable you prepare a favicon.ico file from your existing image or maunally. 
Second add the icon to your Google Site:
  • Starting on the home page of your Google site, 
  • Select  More Actions -> Manage Site -> Attachments
  • Click Upload
  • Browse and select your favicon.ico file
That's all.  Exit management mode and reload your browser page to verify your site's new personalized favicon!

To change or remove your favicon:
  • Select  More Actions -> Manage Site -> Attachments
  • Check the box for the 'favicon.ico' entry and use the Replace or Delete buttons.