I just came acros the following in Genesis:

  if (theme_get_setting('toggle_name') == FALSE) {
    $vars['visibility'] = 'element-invisible';
    $vars['hide_site_name'] = TRUE;
  }
  else {
    $vars['visibility'] = '';
    $vars['hide_site_name'] = FALSE;
  }

Think it could use a bit of inline docs to highlight how to use this and why.

There isn't presently an option in the subtheme admin/appearance/settings/genesis_SUBTHEME to hide site_name as a checkbox, but that is where I would expect to find it.

Also, would suggest that the $visibility variable should be something less generic.