The key is to lie about what is your biggest weakness, while still giving an answer that doesn't come across as a lie nor being dismissive of the interviewer/question. The political tool of answering a related but different question that has a more favorable answer plays well here, such as instead answering "What is one of your weaknesses and how are you overcoming it?"
When asked a "what is your biggest X" kind of question, I almost always wonder aloud if I am the best judge of my own biggest X, whatever X might be. I feel that then gives me freedom to answer the question relative to something that I think is a weakness that would be appropriate to the situation, and how I am addressing it.
For example, talking about being somewhat OCD about things can be considered a weakness. But for some jobs in this field, a little OCD is not necessarily a bad thing.
