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Pindyck, Rubinfeld. Microeconomics (2017 9 ed). pp. 314-315.
Etymonline substantiates that 'rent' originally didn't mean the microeconomics meaning overhead: no surprise, as microeconomics is a recent discipline. So please see the titled question. Why worsen the ambiguity and polysemy of 'rent', rather than using some other word?
Is Economic Rent related to the mainstream meaning of "payment for use of property"?
Note that the Etymoline page you link traces the origin of the word rent as meaning, more generally, "income revenue". Here's their definition (my emphasis):
rent (n.1) "payment for use of property," mid-12c., a legal sense, originally "income, revenue" (late Old English), from Old French rente "payment due; profit, income," from Vulgar Latin *rendita, noun use of fem. past participle of *rendere "to render" (see render (v.)).
This meaning (income revenue) is quite close to the use of the term rent by economists—i.e., an income arising from private control of some scarce factor of production.
Indeed, if we consult the Oxford English Dictionary (which I usually take as the definitive reference on the language), here's what we find in the first entry:
1. b. Revenue, income. Also fig. Obsolete.
The OED has examples of the word being used in this sense as late as 1783. Meanwhile, the first recorded usage of "rent" in the sense meant by economists is 1662. So economists didn't simply invent a new meaning for the word rent; they adopted the word because, at the time early economists were writing, it meant exactly what they wanted to say.
The more modern meaning of "payment for use of property" is essentially a special case of the older, more general meaning of the word rent. Indeed, such a payment is an expense for the payer but an income for the recipient.
If a recipient owns an apartment and earns an income ("rent") from letting it out to a tenant then s/he is earning rent both in the common modern sense of the word, and in the sense described by Pindyck and Rubinfeld. So the usage is clearly related. It's just that this situation (someone earning an income because they own a scarce factor of production) is somewhat more general than the simple contexts in which the word rent finds every day usage.
The three classical factors of production (per Adam Smith, David Ricardo, and their contemporaries) are Land, Labor, and Capital. As discussed on the wikipedia article for land the "income derived from ownership or control of natural resources is referred to as rent." This largely coincides with the intuitive notion of paying rent, hence the name.
Technically, the amount paid by a renter to the owner of land is called "contract rent" and includes not just the economic rent but also a return to capital (interest) on the improvements made to the land (e.g. buildings, infrastructure, etc.) as well as compensation for risk.
For political reasons, other economists attempted to modify these definitions in the late 19th / early 20th centuries, primarily in an attempt to diminish the influence of economist and politician Henry George. Since that time, the role of Land has either been subsumed into Capital, or ignored outright. Alternate definitions of "Economic Rent" have been offered that leave out any reference to Land. This has led directly to the confusion you experience, since it makes the terminology seem arbitrary and/or irrelevant.
Put simply, economic rent is called that because it corresponds to the payment made to the owner of land (or associated natural resources) and modern economic theory has been structured so as to disguise that relationship.
