Drupal - Drupal AJAX fires only once

It's hard to say without actually running those queries but I think the problem is just because of a disparity in the names of your form elements.

In your form the element is defined as $form['date_weekend'] but in your callback you set an element called $form['booking_weekend'].

This would probably be fine except that $form['booking_weekend'] doesn't have the #prefix and #suffix that $form['date_weekend'] does. Drupal replaces the entire wrapper element when it does an AJAX replace so if the new element doesn't have a wrapper with the same ID around it the next AJAX call won't work (as no element would exist with the correct ID).

I'd recommend changing the element names to match as this should solve the problem, but if you really need it to be a different element just add the wrapper around the new element in your callback:

// The rest of delivery_date_weekends() comes before this as normal...

  $form['booking_weekend']['#markup'] = theme('table', array('header' => $headers, 'rows' => $rows, 'sticky' => FALSE));
} else {
    $form['booking_weekend']['#markup'] = 'No delivery weekends exist for this month. Please select another date.';
}

$form['booking_weekend']['#prefix'] = '<div id="dates-ajax">';
$form['booking_weekend']['#suffix'] = '</div>';

return $form['booking_weekend'];