Does dist-configure.php file name need to change?

I am upgrading from 1.3.9b to 1.3.9h, and have noticed the names of the config files in Includes and Admin/Includes have been changed to "dist-configure.php."

Will these files need to be changed to just "configure.php?"

Or are they named this way for a special reason?

No. They're there for a reason.
You need a bit of a history lesson here:

• Zen Cart doesn't come with any "configure.php" files. If it did, then people would accidentally delete their original configure.php files by blindly uploading blank ones from the new version.
• Thus ZC supplies only dist-configure.php files (dist = "distribution"), which are simply stubs that you can rename to configure.php during your FIRST installation so that the installer process can write the collected information into them correctly.
• (The installer can't magically create NEW files, hence the instruction for you to create your own, or simply rename the dist-configure.php files to configure.php to make the process easier.)

If you did an install of ZC via some one-click install provided by your hosting company then you've missed out on all that important learning experience.

The contents of dist-configure.php are really moot, since they're never used. The zc_install installation process completely replaces the contents of configure.php when it is run and told to write out the collected configuration information.