There are some questions your interviewer has no business asking, and this is one.  But while you may feel like answering, "none of your business, "  naturally you can't. Some interviewers ask this question on the chance you admit to something, but if not, at least they'll see how you think on your  feet. 

Some unprepared candidates, flustered by this question,  unburden themselves of guilt from their personal life or career, perhaps expressing regrets regarding a parent, spouse, child, etc. All such answers can be disastrous.

BEST ANSWER:  As with faults and weaknesses,  never confess a regret.  But don't seem as if you're stonewalling either.

Best strategy:  Say you harbor no regrets, then add a principle or habit you practice regularly for healthy human relations.

Example:  Pause for reflection, as if the question never occurred to you.   Then say, "You know, I really can't think of anything."  (Pause again, then add): "I would add that as a general management principle, I've found that the best way to avoid regrets is to avoid causing them in the first place.  I practice one habit that helps me a great deal in this regard.  At the end of each day, I mentally review the day's events and conversations to take a second look at the people and developments I'm involved with and do a doublecheck of what they're likely to be feeling.  Sometimes I'll see things that do need more follow-up, whether a pat on the back, or maybe a five minute chat in someone's office to make sure we're clear on things.whatever."