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.