When Cybermen first appeared, in "The Tenth Planet", several of them did have individual names: Krail, Talon, Shav, Regos Krang (the only one with 2 names given), Jarl & Gern.
Since then, in terms of fully converted Cybermen, none have been identified on screen except for those who have been identified as being formerly Jackie Tyler ("The Age of Steel") and Yvonne Hartman ("Doomsday"). There is also the case of the incomplete conversion of Lisa Hallett in the Torchwood episode "Cyberwoman".
"Handles" in "The Time of the Doctor" was a nickname but it was specific to that individual.
In other media, however, the Doctor Who Magazine comic strip "Throwback: The Soul of a Cyberman" (comic story) featured the adventures of a Cyberman named Kroton.