Brigham Young interprets "Adam" and "Eve" to mean the "first man" and "first woman"; the name "Adam" long predates Adam receiving that name.
Heber C. Kimball's Journal: November 21, 1845 to January 7, 1846 (Salt Lake City, UT: Modern Microfilm Company, n.d.), 124, December 28, 1845
"[i]n the first place the name of man is given a new name, Adam, signifying the first man,—or Eve, the first Woman—Adam's name was more ancient than he was—It was the name of a man long before him, who enjoyed the priesthood. . . . After his fall, another name was give to Adam"