Page Templates – this code only works for one Custom Post Type

Dump the constant MY_CUSTOM_POST_TYPE use the $post->post_type instead.

/**
 * Register the meta box
 */

add_action('add_meta_boxes', 'page_templates_dropdown_metabox');
function page_templates_dropdown_metabox(){
    global $post;
    add_meta_box(
        $post->post_type.'-page-template',
        __('Template', 'my_template'),
        'render_page_template_dropdown_metabox',
        $post->post_type,
        'side', #I prefer placement under the post actions meta box
        'low'
    );
}

/**
 * Render your metabox - This code is similar to what is rendered on the page post type
 * @return void
 */
function render_page_template_dropdown_metabox(){
    global $post;
    $template = get_post_meta($post->ID, '_wp_page_template', true);
    echo "
        <label class="screen-reader-text" for="page_template">Page Template</label>
            <select name="_wp_page_template" id='page_template'>
            <option value="default">Default Template</option>";
            page_template_dropdown($template);
    echo "</select>";
}

/**
 * Save the page template
 * @return void
 */
function save_page_template($post_id){

    # Skip the auto saves
    if ( defined( 'DOING_AUTOSAVE' ) && DOING_AUTOSAVE )
        return;
    elseif ( defined( 'DOING_AJAX' ) && DOING_AJAX )
        return;
    elseif ( defined( 'DOING_CRON' ) && DOING_CRON )
        return;


        update_post_meta($post_id, '_wp_page_template', esc_attr($_POST['_wp_page_template']));
}
add_action('save_post', 'save_page_template');


/**
 * Set the page template
 * @param string $template The determined template from the WordPress brain
 * @return string $template Full path to predefined or custom page template
 */
function set_page_template($template){
    global $post;

        $custom_template = get_post_meta($post->ID, '_wp_page_template', true);
        if($custom_template)
            #since our dropdown only gives the basename, use the locate_template() function to easily find the full path
            return locate_template($custom_template);

    return $template;
}
add_filter('single_template', 'set_page_template');

Code not tested

Code will show the dropdown for all post types.