Can I add Vertical Tab on a Node Edit Page in admin
You can achieve this with hook_form_alter. Similar example can be found in the Examples for Developers module. Take a look at this after you are done with your form tabs, add a custom submit handler like $form[#submit][] = 'my_custom_submit'; and then save the data in submit handler using variable_set() method like this:

variable_set('MYMODULE_' . $form_state['values']['your_value'] . '_special_option', $values);


Note: put it in template.php or in a custom module.