Warning: Selecting fields from a list does NOT work in Acrobat DC due to a bug in the application itself, but the option to automatically rename all duplicate fields will work.

Update (07/08/2019): Added some screenshots of the tool in action.
Also, a warning in advance: Only the option to rename all (duplicate) fields automatically will work in Acrobat DC, due to a bug in that application that prevents the individual fields selection from functioning correctly.

When you use the built-in Duplicate Field command in Acrobat, or even when you just copy & paste a field in your PDF file, the result is a copy of the original, meaning that any changes made to any of the copies will also occur in all of the other ones. This can be useful, but sometimes you want each field to function on its own.
Until now you had to manually rename your duplicated fields in order to achieve this, but this script will do that automatically for you.

New Feature: The script will now maintain the groups of radio-buttons or check-boxes when renaming the fields, including the export values of each widget.
In addition, the script now allows you to easily select multiple fields for renaming, instead of doing it one-by-one.

The tool in action:
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The original file with the duplicate fields and the tool's button on the right

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Clicking the button displays a pop-up menu with several options. We select to rename all fields.

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The resulting file: The duplicated fields have been renamed

Important: When you use this script the renamed fields lost all format/validation/calculation settings that are attached to them.