One approach is to discuss a professional weakness you have been actively working to improve. For example, if you are not the most vigorous writer, you could discuss how you have been taking writing classes or working with a tutor to improve your skills.
Another approach is to choose a personal weakness that will not affect your ability to do the job. For example, you might say that you are a perfectionist or sometimes have trouble letting go of things.
These are some of the traditional approaches and are perfectly valid if you choose to use this to answer this question. But if you want to follow some recent trends, you can carefully evaluate the job description needed to do the job and list the top 5-6 skill sets you need to complete that job well. And brutally honest with yourself and give a rating of 1-10 on how competent you feel you are with those skill sets. Then use that answer to the question.
For example, you can say, I studied the job description, and specific skill sets are critical to doing this job well. Based on my self-evaluation, I feel that my rating is (give yourself a 7 or 8 out of 10). And because this job is so important to me, I have taken extra steps to improve in those areas. For example, (explain what you are doing to improve).
This approach has a few advantages over the more traditional ones.
First, it allows you to avoid any potential pitfalls with the answer. You will not get caught off guard by saying something that could be interpreted as a negative.
Second, it shows that you have done your homework and are serious about the job.
Third, it shows that you are willing to invest in yourself to improve your skills.
Fourth, it allows you to highlight some of your other strengths.
And fifth, it shows that you are proactive and not afraid to take on a challenge.
So, if you want to follow this trend, do your homework and be prepared to discuss your rating honestly. But most importantly, make sure to end on a positive note by discussing what you are doing to improve the situation.