I was recently notified that my 6-month internship with a nonprofit was not approved as part of PSLF because, even though it was full time, 40 hours a week, it was less than 8 months. Where does this requirement come from? All the Department of Education FAQs I’ve seen have stated that employment must be full time, but nothing about a certain length of time. Can this be appealed? If I were to later work for this same employer another 2 months, would I then get credit for the full period?

This is the first time I’ve heard of such a denial, and I believe it is in error.  This is the language from the regulation 34 CFR 685.219:
Full-time (1) means working in qualifying employment in one or more jobs for the greater of—
(i)
(A) An annual average of at least 30 hours per week, or
(B) For a contractual or employment period of at least 8 months, an average of 30 hours per week; or
(ii) Unless the qualifying employment is with two or more employers, the number of hours the employer considers full-time.

I believe your work should be counted as meeting the annual average requirement.  The language about 8 months is intended to ensure that teachers on academic calendars can still get credit for 12 qualifying payments, even if they have summers off.  You shouldn’t get credit for 12 payments, but you should get credit for the 6 payments you made while you were there.

I suggest you start by contacting FedLoan Servicing in writing and ask that this be corrected.  Be in touch with me by email heather[at]askheatherjarvis.com if you need a contact there.