Okay, I figured it out. The following snippet is all that it takes. But thanks again for your effort. You pointed me in the right direction.
<?php
/*
Plugin Name: Show Site Path
Description: Show Site Path in My Sites Menu
*/
function modify_my_site_names( $wp_admin_bar ) {
//get "my sites"
$mysites = $wp_admin_bar->user->{'blogs'};
foreach($mysites as $site) {
$site->blogname .= ' (' . $site->path . ')';
}
}
add_action('admin_bar_menu', 'modify_my_site_names');
?>