The Big Bad Woolf
The play never tells us exactly what the title is supposed to mean, and no one named Virginia Woolf ever shows up within it. This all leads us to ask: just who is this Virginia Woolf person? What does she have to do with the play? Was Albee just being totally random? Probably not. Let's investigate.
Virginia Woolf was a writer famous for her stream of consciousness style. Woolf tried to show the emotional truths churning behind the eyes of her characters; she tried to get inside their heads and really show what it was like to be them. Also, Woolf, like Albee, was a product of the upper class. Her work often criticized and peeled back the layers of pretension that masked her social peers. The fact that Woolf was all about truth and layer peeling, leads some to think that Who's Afraid of Virginia Woolf? is just another way of asking, "Who's afraid to live without illusion?" Albee confirmed this in a Paris Review interview in which he said, "Who's afraid of Virginia Woolf mean who's afraid of the big bad wolf…who's afraid of living without false illusions" (source).
A Bad, Sad Joke
Albee goes on to note that the line, and how it makes a play on Virginia Woolf versus the big bad wolf, seemed to him "a rather typical university, intellectual joke." (source) And as we know from the play George and Martha want to be typical university, intellectual people. Of course, whenever it shows up in the play, the line is used more as ammunition than a funny ha-ha moment. Martha sings it to try and get a rise out of George. Later, George sings it to try and drown out Martha's barbs.
It's, uh, a very dark play, folks.
How Absurd!
We think it's also possible that the meaning of the title might represent a concept on which the play is based: absurdism. Absurdists believe that life has no meaning (at least not one that we can ever be sure of). Therefore, everything we do to create meaning in our lives is ultimately pointless or absurd.
So, what does that have to do with the title? Think about it: the title comes from a joke. It's a parody of "Who's Afraid of the Big Bad Wolf?" from Disney's The Three Little Pigs. Some unknown person sang the parody at the party that the characters attended earlier, and it was apparently hilarious. Notice, though, that Albee never tells us in what context the little ditty was sung. It's like we're getting the punch-line to a joke, but not the set-up. Or maybe, it's the set-up without the punch-line.
In any case, the title is a joke whose meaning the audience doesn't know. The characters are up there laughing it up, while the rest of us are left wondering just what's so funny. Doesn't that kind of sum up the whole absurdist view of life? It's all a big joke, whose meaning is ultimately unknowable.