Not what, but who is Elisa Dolittle? She is a literary character whose story has been made into plays (Pygmalian) films and musicals (My Fair Lady). It is the story of a common London girl working at the Covent Garden market who is 'sold' to Professor Higgins who was a theoretical scientist who was challenged to proove his theory that class and position in life could be taught. He took young Elisa and taught her to speak 'correctly' using elocution techniques (the rain in Spain falls mainly on the plains), dressed her, taught her manners, etc and launched her into high society where she became an exotic mystery, and love interest of a young chap. It may be possible to be refered to as an "Elisa Dolittle" if you are a fake, or are educated above your 'station', or have airs and graces etc. Professor Higgins is another Frankenstein character in that he 'creates' the new Elisa, but doesn't (initially) recognise her as a 'real' person. Elisa herself also identifies with Frankenstein's Creature in that she finds that she cannot fit into either world... the rich world won't accept her, and she cannot return to the world she left behind. Try looking up the film(s) on the web (www.imdb.com) or just googling the play/book.

I've heard the situation where someone uses innapropriately vulgar language or behaviour in a social situation (causing embarrassment to their companions) referred to as "doing an Eliza Doolittle" possibly in reference to the swear word the character uses in the musical "My Fair Lady" when she gets over excited during a day at the races.