Currently it doesn’t seem to exist a filter to manage fields in the menu creation. On the other hand, what you can do is to create a new “menu walker”, to provide WordPress input fields as well as adding your own. That involves the extension of WordPress class Walker_Nav_Menu_Edit with your own code.
Since the explanation of all the needed code is a bit long, I’d like to suggest you to use an existing plugin and use the parts you need. I, for example, used the plugin Menu image as a starting point. Pay attention to the two classes there:
Menu_Image_Plugincreates the basic functionality of the plugin (you will only need some of the filters and actions in the constructor)Menu_Image_Walker_Nav_Menu_Edit, which extendsWalker_Nav_Menu_Edit, to create the fields for the menu
Hope this helps you (and maybe others) to manage their menus the way they want 🙂
UPDATE
As always, while searching for something else, you find an answer for waht you were previously looking for. So.. here is an answer that could help you with a bit of (easy) code; here a more thorough answer, with more (easy-to-medium) code.