How do I alter the options in the add menu link checkbox in the node edit form? I am hooking the node edit form with hook_form_node_page_edit_form_alter(). How can I alter the available options when adding a menu item for that node?

I need to only show menu items of given node ids:

if (isset($form['menu'])) {
  $node_ids = array(1,2,3);
  // only show those node's menu links in the options
}
$form['menu']['link']['menu_parent'] is the form array select box for the menu selector widget. $form['menu']['link']['menu_parent']['#options'] contains the key/value pairs for the menu select box. 

In your given example, you can use unset($form['menu']['link']['menu_parent']['#options']['my_uneeded_menu_id_option']); to remove your  unneeded menu items.