Which FAA certifications may be used as a name suffix and how are they properly used?
I don't know that there's any kind of official rule about what name suffixes a person may use. I suppose as long as he is not being subversive then there's no problem.

These would fall under the "professional" suffix category and like many suffixes may only be recognized by people familiar with a certain industry. For example, CISSP or CISM may only be recognized, ie. they know what it stands for, by those who work in information technology and other related fields.

If it isn't a professional FAA certification, then obviously, it shouldn't be used as a suffix.  I have seen the following used as a suffix or listed as a desirable qualification:


CFI, CFI-I, CFI-G and MEI
AGI, BGI, IGI
ATP, ATP-R
DPE


There is, however, a point that shouldn't be missed;  you won't see pilots going around signing documents and adding suffixes at the end of their name like a doctor, but it is usually a way of conveying to an audience what that person's qualification or source of legitimacy is such as on a business card, advertisement, or flight school staff page.

A commercial pilot would not use that qualification like a suffix because it is not commonly abbreviated, and without any other qualification that would imply that a commercial certificate is held, they are extremely limited on what they can actually get paid to fly for. Therefore, the advertising aspect is irrelevant because there is virtually no public target audience.

The lack of a public advertising target also goes for certificated flight dispatchers.