Correction: Bob Dylan did record the song during the sessions for his first album but it didn't make the cut. This version was released in 1991 on "The Bootleg Series Volumes 1-3." Dylan is only given an arranger's credit. All signs point to this being a traditional song. The question then becomes, how did Dylan learn it?

Some more info from Wikipedia:
"He Was a Friend of Mine is a traditional folk song in which the singer laments the death of a friend. It has been recorded by Bob Dylan, The Byrds, Cat Power, Dave Van Ronk, Mercury Rev and Willie Nelson. The version recorded by Willie Nelson was used in the film Brokeback Mountain and is credited to Dylan, who arranged an early version of the tune in 1962"

"In the Byrds' version, John F. Kennedy was the friend whose death was lamented. The melody is changed considerably and Roger McGuinn takes a songwriter's credit."