In your case I'd probably just hardcode a path, like /usr/share/yourmodule/ and put the data file(s) there. Unless they are pretty small, in which case a __DATA__ section would be even easier. Then change your module to read them when it needs them.
That's the "how" if you don't follow the File::ShareDir route -- though personally I think that even simples modules should have a proper build/install system (via Module::Install or Module::Build), because it's easy to set up and easy to use.
Occasionally certain values will need to be changed, but this would be on an application-wide basis, not on a user-by-user basis.
This indicates that it's not a one-off script, but something that is developed and/or maintained over a longer period of time. I'd definitively go the Module::Install/Module::Build route for that, and use File::ShareDir to locate the data files. But your mileage my vary, of course.