This street names first appears in the 1937 BSD. The entry contains addresses from number 1-17 then the wording for 19-23: "vacant" followed by "vacant ground". It also contains addresses for numbers 2-52. This street is named from its proximity to Ainsworth Street.
It is surmised the "Ainsworth streets" (Ainsworth Avenue/Drive/Parade/Pass/Street) are called after the novelist William Harrison Ainsworth who died in 1882.