By now, you know for certain, that I love eponyms. And this one comes with a "twist" ;D
I would have liked to believe that Elvis Presley's facial hair style would have started the trend but you gotta see the sideburns of Ambrose Burnside to know why they were named after him :D
Ambrose Burnside was a general in theUnion Army in the American Civil War and the guy could have not attained such popularity with his military exploits as he did with his fashion statement. He was the commander of the Army of the Potomac but was relieved of his command after losing the battle of Fredericksburg. His way of wearing his side whiskers along with a moustache but clean shaven chin gave the style the name Burnside's which with time morphed into burnsides and then into sideburns as such a facial pattern was on the sides of a face.
Source: Google, Answers.com