Yeah, what is up with the title? As we have already said elsewhere, Andrew is the play's true protagonist, since it's his story that really drives the plot…so why is it named after his daughter? Well, as we've already noted in "Character Role Identification," the interplay between Andrew's ideas and Barbara's is at the heart of pretty much everything that happens, and dismantling Barbara's idealism is one of Andrew's big victories—one that brings the whole family (minus Lady B) together and indirectly saves his company, since Dolly is then more or less free to take over the Undershaft works.
Because Barbara's transition is really at the heart of all this, we guess it only makes sense that she gets the title.