I cannot figure out how to add the search form to a child element in the menu! I am trying to get something like this http://www.theverge.com/ on my website where you click the search icon and a search bar drops down, but I cannot figure out how to make it work.  I assumed I could do it with menu_build_tree() but have failed quite a bit, so I moved onto THEME_preprocess_menu_link() and have not figured out how to add a child element to a menu item that already exists to print my search form into.

I am pretty new to programming and even newer to Drupal, any suggestions would help me greatly!
The search button on that site looks like it is a menu item however you will find that it is a separate entity, nothing to do with the menu at all.

Usually that kind of thing is achieved using Javascript and AJAX.  Javascript to make the form appear, and AJAX to submit it.

Have a look at the Popup module, it might be what you are after.