Make a block from a "part" of a drop down menu
I would advise using menu block (https://drupal.org/project/menu_block), which allows blocks to be created from menus. The most important part would probably be starting level and max depth, which allows you to specify the options you'd like to use.

If you'd like an example usage similar to what you're attempting to achieve, just let me know and I'll take some screenshots.