child_of not working in wp_dropdown_categories

The variable $book_term_id has not been set. You either need to pass it to your function or set it in your function.

Pass it to your function:

<?php dropdown($book_term_id); ?>

Set it in your function:

function dropdown(){
  $book_term_id = 5; // if a global variable, you can just use:  global $book_term_id;  
  $dropdown = array(
    'child_of'   => $book_term_id,
    'hide_empty' => 0
  )
  wp_dropdown_categories( $dropdown );
}