If they choose a question they were hoping for, this lets them talk about something they think is important and which shouldn't be missed given the limited time remaining. Interviews should be a two way thing, not them just answering your questions.
If they choose a question they were dreading, this lets them demonstrate their honesty/bravery/confidence and their desire to build mutual trust by sharing a question that they may not have a good answer for, and that they are not just sell themselves to you.
It's good because there is no right or wrong answer. As the interviewer you can't fall into the trap of comparing the answer to any predetermined answer you might have had in mind. I really like trying to keep an open mind, not match a candidate to some predetermined ideal I might have had.
Finally, it shows the candidate that you value their input into a discussion, and good candidates are interviewing you as much as you are interviewing them. Showing your value their input, not just that you want to see if they can pass your test, can only increase their interest in the role.
I started this list after seeing this tweet by Sascha Kerbert. I knew I would never find that thread again, and I wanted to keep all of the replies to help in future interviews.
I decided to share it and see if we can gather more ideas.
So, what's your favourite interview question?