Using a multiple element in widget form

I would remove the jQuery save function that was added… and then…

Try this for the form function:

public function form( $instance ) {
    // Code for editing/adding title and adding IDs to the <select> object
    // ...

    <?php
            printf (
                '<select multiple="multiple" name="%s[]" id="%s" class="widefat" size="15" style="margin-bottom:10px">',
                $this->get_field_name('ID_list'),
                $this->get_field_id('ID_list')
            );

            // Each individual option
            foreach( $instance['ID_list'] as $id )
            {
                printf(
                    '<option value="%s" %s style="margin-bottom:3px;">%s</option>',
                    $id,
                    in_array( $id, $instance['ID_list']) ? 'selected="selected"' : '',
                    $id
                );
            }

            echo '</select>';
}

Alright, try this update function:

   <?php
    function update( $new_instance, $old_instance ) {
        $instance = $old_instance;
        $instance['ID_list'] = esc_sql( $new_instance['ID_list'] );
        $instance['title'] = esc_sql( $new_instance['title']);
        return $instance;
    }