There are two traps in answering this question. The first is to say I do not have any weaknesses, or I cannot think of any. The reality is we all have weaknesses and chances are if you do not identify any, the interview may very likely be cut short. The second is not being prepared to answer this question. Make sure you are prepared to identify up to two weaknesses.
There are two keys to nailing this question. The first is to identify a weakness that will not compromise your competency in the role you are seeking and the second is to identify what you are doing to correct the weakness.
As an example, you are currently an Accountant with the ambition to become a Controller at some point in your career. You are applying for an accounting role in a large organization and you know there is room to advance your career in this organization. The weakness you have identified to get from one role to the next is speaking in public and experience and exposure to leadership opportunities. So many of us look or wait for opportunities within our current companies and that just is not good enough anymore. We need to be very proactive about our career and gain the experience externally as well.
So now we get to the big question and your response could be One of my goals is to become a Controller. I know I need to have strong communication and leadership skills and in preparation I have joined Toastmasters. I am learning different skills with every speech and I am also learning how to present different material to different audiences. I also volunteer for a variety of club roles every week which is building on my leadership ability. Within 10 months, I plan on having 10 speeches completed which will give me my Competent Communicator designation within Toastmasters. Within 6 months after that my goal is to have my Competent Leader designation which includes taking on over 15 different leadership roles.
This response would assume that you have taken a good hard look at where you are in your career and where you want to be in the future. Never say something that is not true. The response does not compromise the competency of an Accountant as at the beginning levels of this career, making presentations is not standard practice, however, the more senior you get in the Accounting function, not only will reports and presentations be required but leadership skills whether formal with staff reporting into you or informal with training and guiding new Accountants will be a required skill. This response also indicates what you are doing about your weakness so you have nailed both keys to this question with a response like this.
Since we are in the era of lifelong learning you are letting the interviewer know that you believe in this concept, you are also demonstrating initiative and accountability; all great character traits of a future leader.
On a side note, I highly recommend Toastmasters for all my clients that I coach for so many reasons. Communication and leadership skills are becoming the most critical skills anyone can have, whether it is for your career or personal life. I encourage you to find a club that suits your needs. Whether it is for communication and leadership, significantly improving your self esteem, networking to meet other people, or just wanting to learn about so many things in a relaxed friendly environment. The right Toastmasters club can do all of that. It may feel very intimidating walking through the door for the first time but in order to break through your weakness or fears, the first step is always the hardest. Toastmaster clubs are designed to be warm and inviting. You can attend any Toastmasters club 3 times before you will be asked to join so that means you can try a few clubs a few times before you decide what fits your needs, goals and personality the best. Remember when you hear the great speakers at the clubs you attend, do not be intimidated thinking they are way better than you. Just ask them how they were when they first joined and they will tell you how nervous they were when they started.
In this time when so many of us are very price conscious, the annual membership fees are very reasonable and when you stick with it you will get way more than you anticipate. The encouragement and confidence you will gain will be outstanding. The friends you will make will be an additional blessing too.
In closing, spend some time determining your weaknesses, especially if they are becoming blockers in your life. Determine what support you need to change them and move in that direction. Change does not happen overnight but it will happen with perseverance, so set a long term goal and break it into smaller pieces so you can see your improvements.