Why get rid of the follow links in the weblinks?
Well, page rank is one of the barebones of google and if you want to strike on the top you want to get a high page rank. And if you have alot of weblinks your page rank will "leak" to these sites. Rel="nofollow" is the key here. We have been looking for a good module/plugin/componen to fix the leaking of page rank, but with no result. It seems like the only way is to hack joomla in order to fix this.

Go to com_weblinks folder and edit weblinks.html.php

Change this code:
// open in a new window
               $item->link = '<a href="'. $link .'" target="_blank"  class="'. $menuclass .'">'. $this->escape($item->title) .'</a>';

To this code:
// open in a new window
               $item->link = '<a href="'. $link .'" target="_blank" rel="nofollow" class="'. $menuclass .'">'. $this->escape($item->title) .'</a>';

If you set your links to open in new browser they will now be no follow.