The "tell me one of your weaknesses" question is a classic stress question. It basically asks you to reveal something negative about yourself during a process where you hope to reveal only positives. It forces you to pick out something about yourself that reveals your humanity but doesn't downgrade your attractiveness to the program.
You can try to play the game by giving an example of something that is on the face a supposed negative, but really a positive, i.e., "I'm a perfectionist", "I care too much", "I'm a workaholic," blah, blah, blah. These answers are trite and pretty obvious and will neither ding you or make you memorable.
Another tactic is to bring up a *real* weakness, i.e., "I have anger issues", "I'm a former alcoholic", "I gamble a lot of money". There is greater risk/reward with this tactic, as you not only reveal a *real* weakness, but also give yourself the opportunity to state how you are dealing with, or have overcome that weakness. Of course, if you mention small farm animals, it doesn't matter how well you're compensating for it. You're cooked.
You'll encounter this question in residency interviews as well, as I've encountered, and quite honestly, it's a crock.
Bottom line, they want to know if you are aware of your shortcomings, and whether you are dealing with them. They want to know that *you* know that you're only human, and not an infallable machine. It's okay to have weaknesses, it's great if you know them and work on them. It's unforgiveable to deny you have them.
So this is what I've been doing during *this* round of interviews. When asked this question, I take a moment, then say "I recognize this as a stress question. And while I can point out any one of the weaknesses I have, it would be more important to see the point behind that question. I am human, I am not infalliable. I know what my weaknesses are and try to be aware and work on them everyday."
That has stunned one or two interviewers, and they usually say something like "That's a lot better answer than 'I'm a perfectionist.' That one is overused and quite frankly, crappy."