It has been asked by @Lnafziger for authority supporting an answer I have given to the question. Since this answer is not what I would call a succinct answer, I have made it separate from my initial answer.
14 CFR 61.113(c) A private pilot may not pay less than the pro rata
share of the operating expenses of a flight with passengers, provided
the expenses involve only fuel, oil, airport expenditures, or rental
The regulation by context refers to operating expenses of a flight. The OP asks about a passenger headset rental. The regulation addresses expenses of flight.
Consider if the headset was bought by the pilot and transferred to the passenger friend, with the passenger friend reimbursing the pilot for the cost. Would that be an expense of flight? No, it is not aircraft rental, and furthermore it is not a requirement for flight. So avoid confusing a FBO rental of a headset with a rental of an aircraft.
In the Sommer LOI by FAA Counsel, 61.113(a) is referenced, as prohibiting the "compensation or hire". If our friend were to be given the purchased headset at the completion of the flight, that would be compensation or hire. In the LOI the FAA makes clear their long standing position that compensation or hire can include any benefit or consideration, even if there is no payment. But in the OP example, the headset was merely rented, and solely for the benefit of the passenger, and was not needed for flight as it was for personal comfort or hearing protection. Most importantly, there is no benefit to the pilot.
In the same LOI, the FAA touches upon expenses of flight not covered by 61.113(c), and while they are not enumerated, nor are examples given, the FAA makes clear that 61.113(c) applies specifically to the expenses enumerated in 61.113(c).
Sommer LOI: https://www.faa.gov/about/office_org/headquarters_offices/agc/pol_adjudication/agc200/interpretations/data/interps/2010/sommer%20-%20%282010%29%20legal%20interpretation.pdf
In 1997 the FAA, in Harrington, made a stance that they would interpret compensation and hire broadly. In subsequent interpretations and enforcement actions, compensation and hire were liberally construed. In the instant case presented by the OP, even liberally construed, the rental of a headset provides no compensation to the pilot, is not necessary under regulations for the flight, and is not a fuel, oil, airport or [aircraft] rental fee.
[There are other LOI which more specifically address scenarios virtually identical to the OP's question, however, those LOIs were not found online. Sommer is more current, and touches upon the key points which need to be considered.]
As a final note, it is important when reading regulations to interpret just the regulation within the context of the body of regulation. Stated differently, avoid reading into one's interpretation issues which are separate and distinct from the body of regulation. Furthermore, it is noteworthy that while FAA LOIs are not regulatory they provide insight as to how the FAA might deal with a matter, or might be persuaded to deal with a matter. (Having said that, I know of cases where they appear to attempt to enforce regulations which are not codified. Yes, the pilot can appeal to the NTSB, but a friend faced with that gave up when the FAA said at the end of highly politicized first day of the hearing that they estimated 23 days of material. His attorney advised him he had just spent $13k on the first day, and he decided to fold. An adversary is powerful armed with deep pockets.)