"... probably named after Lady Elizabeth, daughter of the First Marquis of Donegall, with Charlotte Street, Amelia Street and Arthur Street being named after her siblings" (Patton 1993, p. 144). Henrietta Street can be added to this list. Lady Elizabeth Juliana Chichester (1767-87) was the only daughter of Lord Donegall and Lady Anne who attained adulthood.
Eliza Street was first recorded c. 1830 on the 1st edition 6" OS map; Martin's Directory, 1839.