Endorphin means "the morphine inside you." The word was built to describe a guess that turned out to be right: if the brain responds to opium drugs, it probably makes something similar of its own.
By the 1950s scientists knew that opiates work by docking onto receptors on nerve cells. That raised an obvious question. Why would the brain carry receptors for a poppy extract unless it had a natural chemical that fit them?
The hunt was messy. In Aberdeen, the pharmacologist John Hughes ground up pig brains, which he got from slaughterhouse workers, according to the Science History Institute, by bribing them with scotch. His mentor Hans Kosterlitz tested the extracts on tissues that stop contracting in the presence of morphine-like chemicals. Something in the brain extracts did exactly that. Hughes and Kosterlitz announced their results in May 1974 and published the structures of two small peptides in Nature in December 1975. Those were named enkephalins, from Greek enkephalos, "brain," literally "in the head."
"Endorphin" became the wider name for the brain's own opiates. It joins endo-, from Greek endon, "within," taken here through "endogenous," with "morphine." Etymonline records it from 1975, borrowed from French endorphine. Merriam-Webster gives a first known use of 1976. A University of Michigan archive credits the American pharmacologist Eric Simon with coining the word, but it gives no date.
The second half has its own story. Morphine was named in 1816 by the German apothecary Friedrich Sertürner. He alluded to Morpheus, the god of dreams, because the drug brings sleep. So endorphin carries Morpheus inside a modern lab word.
The popular link between endorphins and the "runner's high" came later. It is a reasonable idea but not a settled one. Encyclopedia.com says only that endorphins "may be" involved, noting that blood levels of beta-endorphin rise during exercise.