This is a hidden option of Google sites, but it is very simple operation.
I've used this method to put the
icon you can see at your browser address line.
Have your favicon ready on your computer (you can find free favicon on the web or create your own).
Log into your Google site.
Click the 'More' button at the upper right corner of the screen,
and select the 'Manage Site' option.
On the right menu list, select the 'Attachments' option.
Pres the 'Upload' button on the upper menu of the screen.
In the popup window, navigate to the location where your favicon is saved and select it, Press 'Open'.
In order for this to work, the file name MUST be named 'favicon.ico'.
In case the file name was different, you can mark the check box of the relevant file in the list, and press the 'Rename' button in the upper menu of the screen.
The file must also be located at the root location of your site, which is denoted as '/' at the location column of the file list.
In case the file is not locate in the root location, you can again mark the checkbox, and choose the 'Move' option from the upper menu of the screen and move the file to the proper location.