Emil, if I understand you correctly you would like to set up favicon only for the mobile version of your site. Or set different favicons for Desktop version and mobile version.
In this case you will need to edit ow_themes/origin/mobile/master_pages/mobile_html_document.html file. Paste before </head> the following code:
<link href="http://yoursite.com/ow_userfiles/plugins/base/favicon.ico" type="image/x-icon" rel="shortcut icon">
So, the result will be:
<link href="http://yoursite.com/ow_userfiles/plugins/base/favicon.ico" type="image/x-icon" rel="shortcut icon">
</head>
Save the changes and refresh the browser cache. So, when you click the "Mobile version" link, you will see the new added favicon.
Be note the favicon displaying depends on the browser type and its cache settings.
For checking the changes press F12 button and find new code line in the header section of your site.
If you still have issues, repeat this actions on the default Oxwall theme- Origin. We've checked it and everything is working.
Ensure you have updated your software.