If my son got arrested for shoplifting 9 years ago and he is now in nursing school and this showed up on a background check, how does he clear his record?

In Massachusetts, your son would have to file a petition to seal with either the court that the charges were brought in or the board of probation, depending on several factors. 

Another possibility is to file a motion to reopen the case and to enter a dismissal. 

He should contact a local attorney to discuss what the best option is to move forward.