Allows you to search by the payment ID on the downloads > payment history screen. Also allows you to search on the downloads page if you know the ID.

gistfile1.php

<?php

function edd_search_by_id( $where ) {
	if( is_admin() ) {
		global $wpdb, $pagenow, $typenow;
		
		if ( ! ( 'download' == $typenow && in_array( $pagenow, array( 'edit.php' ) ) ) )
			return $where;

		if ( isset( $_GET['s'] ) && !empty( $_GET['s'] ) && intval( $_GET['s'] ) != 0 ) {
			$where .= ' or ' . $wpdb->posts . '.ID = ' . $_GET['s'];
		}
	}
	return $where;
}
add_filter( 'posts_where', 'edd_search_by_id' );