I’d like to copy and re-name that piece of code that begins
class contact_info_widget extends WP_Widget {

Duplicate it somewhere in the child theme and re-name and extend it and have it get initialized and see it in the Widgets section as a new widget. Do I create a new functions.php file in the child theme to do this? Can I use an includes/my_widgets.php file in the child?

Sure, sounds like you want to declare your own class, then add the new features to it!
So make sure you define your own class … for example contact_info_widget2 along with the hook that registers it: register_widget(‘contact_info_widget2′);

in your child theme’s functions.php or in some file included by it.
Then in the class itself you can modify/add whatever you need!