I had never heard of a free application for providing live support / live chat on a website.

One of my friends (Eric) mentioned this the other day. It is called Crafty Syntax and I am posting the link below:
http://www.craftysyntax.com/

You can use this to add live support to your website free of charge.

Also, if your site is run on Drupal, then you can refer to the below page to see how Crafty Syntax can be integrated into Drupal.
http://drupal.org/node/248920

As always, I am excited seeing free and open source. But if you want to support them, you can always donate.