How can I handle this sort of question:
Give an example of when you had to change the way you did something in order to solve a problem?
Note: I am posting the interview questions here because of answers here.
Using someone else's answer to this question is completely useless. All you will learn or demonstrate is that someone else is a good or bad fit for that job. What you need to do is learn how to find your own answer to this question. As Ross says, this happens dozens of times a day.
If you're in "interview mode" what you want to do is come up with examples of this type that are larger, and that are relevant to the job you're applying for. That time the server was down but you needed to [something] so you [something'ed] instead. That time your boss was rushed to the hospital and couldn't approve the deploy scheduled for that afternoon, so you [whatever'ed] instead. It's ok to make yourself the hero in this story - that's part of the point of it. You can even practice telling the story a few times so that you get things in the right order.
When I'm interviewing you, and I hear this story, you want me to think "this is the kind of person we need! Resourceful, imaginative, and yet respectful of the motivations behind the process." You don't want me to think "is that all you've got?" or "oh no, one of those cowboys who can't wait to abandon all the process because it's supposedly an emergency." Or, if the story is about getting a person to help you, again you want me thinking "yes, I can see what a benefit it would be to have this candidate on our team" not "oh lord I cannot let this one near Steve it would be a fistfight within a week". So make sure your story tells the truth about who you are and how you work.
I see this as a question of how creative, adaptable and persistent someone is in handling the case of what does work versus what should work. There can be steps taken to solve a problem that in some cases may need to be handled differently.
To give an example from Software Development in regards to fixing a bug. If the bug is found in a testing environment, I may have a process of reproducing the bug, looking for various solutions, making a short list of options that I hand to a manager to determine which makes the most sense from a business perspective and then implement the fix, test it, get a code review and then commit to the version control for the next push. However, if there is a bug found in production that has to be fixed immediately, I may change this approach as I may not have time to do things with so much bureaucracy to it. I may jump into finding the bug, changing the code, get it reviewed and then pushed into production in an expedited way.
The question has a few pieces to consider as there is something to be said for how big of a change is there, what caused the change to be done and what was learned from this new approach.
I've had loads of these questions and I find them close to useless in determining anything but the interviewees ability to speak.
Constantly throughout the day and throughout my working life I "change the way [I do] something in order to solve a problem", this is the nature of problem solving and adaptability. However, as this is a normal and frequent occurrence, I take very little notice of it so when asked a question like this, I find it easier just to make up a nice sounding story (with some basis in truth and without embellishing your skills above your ability) if nothing pops immediately to mind, than sit and try think of a specific occurrence.
Normally I would feel guilty about lying in an interview but what benefit can an interviewer get from an answer to this question? Who is going to say "I haven't"? no one. So those who have never been in this situation (or can't think of one) are going to lie and there's no way to validate the tale told. Those who have are going to tell you a story very much like not only the person who lies, but like every other interviewee.
So in short, if nothing pops into mind give an example of a generic situation in which you should constantly find yourself.