How to remove the default "Anonymous" from comment author name? In the comment form field (core comment module) I would to leave the comment author name empty, in order to encourage commentators to come up with original epithets. 

What's the easiest way to do it?
hook_form_alter()

http://api.drupal.org/api/drupal/modules--system--system.api.php/function/hook_form_alter/7

Allows you to hook in to and alter any form, including setting the default text to empty. Here's some more help on what the valid elements are:

http://api.drupal.org/api/drupal/developer--topics--forms_api_reference.html

Best way to start is to determine the form ID by creating a mymodule_form_alter() function and printing out the $form_id using print so it appears at the top of the page. (Or better still, use the devel module to print it out in a Drupal status message). Once you know the form ID, you can add an if to make sure you only act on that form and then alter the array to remove the default value you don't want.

Look at pretty much any module for an example implementation. Many modules use this hook. I can't think of any specific example off the top of my head, but there are loads.

EDIT: Here's a good Lullabot blog post on the matter... there's not much different between Drupal 6 and Drupal 7 in this regard, AFAIK:
http://www.lullabot.com/articles/modifying-forms-drupal-5-and-6