The term side burns comes from Major General Ambrose E Burnside!

Ambrose E Burnside was a Major general during the Early 1800’s. In fact, at the outbreak of the Civil War, he organized the 1st Rhode Island Infantry, which was one of the first units to arrive in Washington and offer capitol protection.

After the war, he briefly serve as the Senator from the state of Rhodes Island. In addition to his many other accomplishments, Burnside has some impressive facial hair. His moustache seemed to reach all the way to the back of his head.

Does this sound familiar? In fact, his facial hair often identified him, as he wore it for most of his life. Soon, the style began to catch on, and took the title of side burns after his last name. 

(Source)