The origin of Elizabeth street's name is unknown, although the prevailing hypothesis for late-19th / early 20th century street names is, by default, daughter of a rich guy.
It first appears on a 1907 sketch map running from North Durham south to Ramseur Street. North Elizabeth was significantly realigned south of Holloway Street in the late 1960s; the buildings west of the Norfolk Southern spur line were demolished, including everything on North Elizabeth Street south of Holloway Street. The road alignment was then moved eastward, adjacent to the spur line, to connect with Fayetteville Street south of East Main Street.
South Elizabeth Street remains along the previous road alignment, meaning that North Elizabeth is continuous with Fayetteville Street, but not with South Elizabeth Street, which is ~ one block west.