A placebo is a fake pill, and the word began as the first line of a funeral service.
Placebo is the Latin future tense of placere, "to please," so it means "I shall please." It opens a line from the Vulgate Latin Bible, Placebo Domino in regione vivorum, usually rendered "I will please the Lord in the land of the living." In the Vulgate numbering this is Psalm 116:9.
That line became the first antiphon of Vespers in the Office of the Dead, the sung evening service for the departed. By the early 13th century, English speakers were calling the whole opening rite "placebo," after its first word. The same habit gave English dirge, from dirige, the first word of a different antiphon in the same service, which meant "direct."
The medical sense is much later. Etymonline records placebo by 1785 for a medicine given to please the patient rather than to help them. By 1811 it was defined as any medicine "adapted more to please than to benefit the patient."
The phrase placebo effect is first recorded in 1900. The idea that patients improve because they expect to improve gained scientific weight in 1955, when a study titled "The Powerful Placebo" argued that placebo responses mattered clinically. A reassessment in 1997 found that the study had not accounted for regression to the mean, and that its data showed no placebo effect once that was done.
That makes the word's journey unusually tidy. A sung promise to please God turned into a pill designed to please the patient. The original meaning stayed attached the whole way.
One detail worth flagging: you will often read that placebo referred to professional mourners paid to sing at funerals. The sources used here confirm the funeral setting and the psalm, but not the paid mourners, so treat that part as an unproven embellishment.