@tarotmeister
The minimum you should have to do for a child theme to show up is:
1: navigate to the themes folder in the wp-content folder on your godaddy file manager
2: inside that folder, twentyeleven and twentyten I believe should already be there. You would make a new folder/directory in the themes folder, alongside the other themes named whatever you want
3: inside your new folder you would make a new file named style.css with the very basic header elements, and possibly the @import rule if needed
If you have done those steps, you should have a theme to activate from within the WP dashboard
as @digital Raindrops said you could upload one of their basic child themes to get started. And also, activating a broken theme could cause problems, as could messing with a live theme. However now that you are familiar with godaddy’s file manager, fixing the problem is as simple as deleting the broken theme.