Favicon is a small icon shown in your browser in front of the URL.
By default Moodle has it’s own favicon with the default theme. To replace the original favicon.ico with your own, upload it under “/home/user/public_html/theme/standard/pix”. If you are using a theme other than the standard one, you have to upload it under the directory corresponding to the theme you have chosen.
The new favicon may not be noticable because the image may be in the browser cache. Moodle cached the old one.
Access Settings > Appearance > Themes > Theme settings > Theme designer mode check box and save. Refresh your browser. Uncheck the box after it works.
If you still face the same issue, find for a file which has content as below
<link rel=”shortcut icon” href=
Replace it with the absolute path OR the URL to the favicon.ico file.
<link href=”/home/user/public_html/theme/standard/pix/favicon.ico” rel=”shortcut icon” />
<link href=”http://yourmoodle.com/theme/new/favicon.ico” rel=”shortcut icon” />
Browse through Favicon Wiki to know more about how exactly favicon works & utilities using which you can create favicon online.