I would say the best way to answer the question is to answer it honestly. The last thing you need to be doing is trying to make someone believe you are good at something that is actually your weakness. If they won't hire you because you were honest then they simply didn't want you and that doesn't mean there was anything wrong with you. Best to just search for the job that wants you as you are.
I have one that I have used many times... and it's the truth too.
Years ago, when I first started in the working world, I worked at a Savings and Loan. The manager there gave me my first annual review and she gave me a lower score for being to organized! Can you believe that! She said I was to organized and it made her crazy because she wasn't organized at all.
So anytime I am asked that question, I just tell them how I got a point off for being to organized. So, I considered it as a weakness... but one I wasn't willing to give up.
It worked out good for me in all jobs I had after that. People just couldn't understand how someone can be so organized... they all LOVE having me around to help them organized themselves. LOL
And therein lies the problem with the question: Everyone has their own idea of what a weakness is so what is a strength to one is a weakness to another. Making hiring decisions based on the opinions of anyone is foolish. Perhaps that interviewer should deal with their own sloppiness instead of expecting the organized to be sloppy so they don't have to be bothered by it?