There wasn't a DOOM difficulty names mod updated for Mayhem and One Down, so I made one myself. Also includes DOOM 64 and Zandronum modes as a bonus. The difficulty names, in order, are listed below.

Classic
I'm too young to die
I'm a wimp
Hey, not too rough
Hurt me plenty
Ultra-Violence
Nightmare!
Ultra-Nightmare!!

DOOM 64
I'm scared!
Be gentle!
Bring it on!
I own doom!
I'm slightly masochistic!
Watch me die!
Hardcore!

Zandronum/Skulltag
I want my mommy!
Pansy
I'm allergic to pain
Bring it on
I thrive off pain
Nightmare!
Ultra-Nightmare!!

Installation instructions
Place ONLY ONE folder of your choice in your mods folder, they will be named accordingly.