If you're using Roundcube 1.4, you can specify the URL to your new favicon in the Roundcube config file (config/config.inc.php.) Simply add this code at the end of the file:

$config['skin_logo'] = [
    '[favicon]' => 'https://your-server.com/favicon.ico',
];

(Of course, change the URL to point to your custom icon image.)

If you're using Roundcube 1.3, you'll need to replace the existing file on the server, since this older version of Roundcube doesn't support setting favicon in the config.

All our skins on Roundcube 1.3 use the icon file located in the larry skin folder, so you'll need to replace this file:

skins/larry/images/favicon.ico