function YOURTHEME_preprocess_page(&$variables, $hook) {
  if (theme_get_setting('toggle_name')) {
    $path = drupal_get_path_alias();
    if (drupal_match_path($path, 'one*')) {
      $variables['site_name'] = filter_xss_admin('Site Name One');
    } else if (drupal_match_path($path, 'two*')) {
      $variables['site_name'] = filter_xss_admin('Site Name Two');
    } else if (drupal_match_path($path, 'three*')) {
      $variables['site_name'] = filter_xss_admin('Site Name Three');
    }
  }
}

simple preprocess function to alter the site name depending on the section of the site

For Changing Drupal Site Name on URL Alias change

drupal

PHP