Thanks to @benjifisher, I found the help for :browse
which shows how to do what I wanted.
More specifically, the default filetype plugin for C/C++ contains these lines:
let b:browsefilter = "C++ Source Files (*.cpp *.c++)\t*.cpp;*.c++\n" .
\ "C Header Files (*.h)\t*.h\n" .
\ "C Source Files (*.c)\t*.c\n" .
\ "All Files (*.*)\t*.*\n"
I copied the file into my local vim ftplugin
directory and modified to my liking which now takes precedence over the system version.