That's really just a small trick, add this code... somewhere, for example in Load.php, close to:
call_integration_hook('integrate_load_theme');
Yes, I'm just being lazy, I don't want to write the package for it. 
I created the package and uploaded as usual at github, scroll down for details.
addInlineJavascript('
$(document).ready(function() {
$(\'.bbc_link\').each(function() {
var url = $(this).attr(\'href\'),
domain = url.match(/:\/\/(.[^/]+)/)[1],
schema = url.match(/^(http[s]*):\/\//)[1];
$(this).css({
\'background-image\': \'url(//www.google.com/s2/favicons?domain=\' + schema + \'://\' + domain + \')\',
\'background-repeat\': \'no-repeat\',
\'padding-left\': \'20px\'
});
});
})', true);
This will show the favicon of the site you linked in posts.
So, using for example:
[url=http://www.elkarte.net/community/index.php]ElkArte[/url]
It will show:
ElkArte
License
o This ElkArte Addon is subject to the terms of the BSD "3-clause" license. You can obtain a copy of the License at http://opensource.org/licenses/BSD-3-Clause
Repository / Download
Change log