Doing a hook_form_alter on the node edit form should do the trick:
use Drupal\Core\Form\FormStateInterface;
function MYMODULE_form_alter(&$form, FormStateInterface $form_state, $form_id) {
/** to appear on the node/add/page form, change this or add more for other content types **/
if ($form_id == 'node_page_form') {
$menuItems = $form['menu']['link']['menu_parent']['#options'];
$menuItemsWithWeight = array();
foreach ($menuItems as $menuKey => $menuValue) {
$storage = Drupal::entityManager()->getStorage('menu_link_content');
$uuid = ltrim(strstr($menuKey, ':'), ':');
$uuid = ltrim(strstr($uuid, ':'), ':');
$link = $storage->loadByProperties(['uuid' => $uuid]);
if(count($link) > 0) {
$link = reset($link);
$weight = $link->getWeight();
$menuValue = $menuValue . ' ('.$weight.')';
}
$menuItemsWithWeight[$menuKey] = $menuValue;
}
$form['menu']['link']['menu_parent']['#options'] = $menuItemsWithWeight;
}
}