Dude is not the name for an ingrown hair on an elephant. Ingrown hair on any human or animal is known by the medical term pseudofolliculitis, which denotes a hair that grows through the wall of a hair follicle or under the skin surface instead of through the follicle opening.
According to Etymononline, dude originated in the late 1800s as a slang term referencing a fastidious man. This term was later used in reference to city dwellers vacationing in the West at what became known as dude ranches. There are various urban legends suggesting that it names ingrown hair on elephants, cows or horses, but they have no basis in fact.