Whenever you produce a custom theme in WordPress, you can include a hyperlink for your theme choices on the left menu within the Wordpress dashboard. The default icon that's used alongside your menu label is known as generic.png and resides within the wordpress-admin/images directory.

Anybody understand how to tell Wordpress to make use of my theme's custom favicon.png rather than the default?

I believed that one out. Hopefully it can benefit others:

add_menu_page('Page title', 'Top-level menu title', 
'administrator', 'my-top-level-handle',
'my_magic_function','../wp-content/themes/yourTheme/img/favicon.png');

You may also give a submenu beneath your theme's primary food selection like so...

add_submenu_page( 'my-top-level-handle', 'Page title',
'Sub-menu title', 'administrator', 'my-submenu-handle', 'my_magic_function')