Display another field as 'Node author' if node posted anonymously
You have two viable options:


In theme system, use either node.tpl.php file to do it, or probably better template_preprocess_node(&amp;$variables). I'll discuss function as that's what I prefer. When using function, copy it to your own template.php, renaming to mytheme_preprocess_node(&amp;$variables). As a source, use parent theme if it implements it. If not, or if there is no parent theme, use link I provided above.

Then find a line like this:

$variables['node'] = $variables['elements']['#node'];


and after it modify $variables['node'] as you see fit. 
Use Computed Field to create a field that shows author's name, and in case of anonymous authors displays arbitrary text. Turn off displaying author and use this field instead.