ok, it's hard to say why that module isn't showing you the full menus. But so long as you can see menu on other modules then there must be a way to sort it out.

Try making a new page and add the links module to it to ensure that it does in fact work normally elsewhere.

 Then use the iconbar function that lets you insert a module from another page, to retrieve a copy of the faulty module to your new page. See if it lets you access the module settings there.

If it does look like it is just that one instance of the module that is faulty, then you could duplicate the hompage without that module and then point the portal to the new homepage in the site settings. Then delete the old page. (Don't do anything I tell you to do here without backing up first - I dont' want to come back here tomorrow and find it's all broken.)

Or get hold of the guy who made the site and ask him for advise.

Or get into the database and get your hands dirty if you know your way around in there.

Rob