Could ADA not be a short for ADELAIDE? In ancient Greece there was an exact name, ADA, she was the daughter of EkAtomnos of Karia.
("Thesaurus of Greek Names", Ares Diamantes, 17.800 names).
The LGPN ("Lexicon of the Greek Personal Names", over 28.000 names,
www.lgpn.ox.ac.uk),mentions 15 different people with the name ADA in the ancient Greek onomatology.