Simply put, there are no standard required qualifications. Qualifications can and do vary drastically from company to company, and even within organizations.
Whereas some professions like engineers or accountants usually have degree, testing, or experience requirements, an HR representative who is investigating a workplace complaint can be an employment attorney, an industrial/organizational psychologist like myself, or a career HR professional. Sometimes they will have HR certifications and formal training, and sometimes not, or even the guy in payroll will double duty as HR when necessary. Those are just examples. I've seen them all.
The very worst I have seen are the ones who have absolutely no background in employment law or HR (and perhaps little interest, too) yet have just been moved into the position as a rotational development assignment with only learn-as-you-go training. Yes, your investigator may simply be someone whom management thought was a good communicator!
When you're talking about alleged harassment, discrimination, fraud/theft, stalking and violence in the workplace, and other serious issues that can impact your livelihood and your daily life, how good does that make you feel?
As a general rule, it's best to know who is in the HR Investigations role before you ever have to use their services.