You can use CopyToClipboard
:
Button["display text", CopyToClipboard["file name"], Appearance -> None]Frameless]
"Copying a string copies the contents of the string."
Column[MapThread[ButtonColumn @ #MapThread[Button[#, CopyToClipboard @ ##2, Appearance -> None]&Frameless] &,
{{"display text 1", "display text 2"}, {"file name", "Alex E"}}]