The first eavesdroppers got wet. They stood in the drip line under a roof, close enough to a wall or window to hear what was said inside.
The word starts with eaves, the overhanging edge of a roof. Etymonline gives the Old English form as efes. Word Histories points out that it was singular. The final "s" was later mistaken for a plural ending, which is why we now say "the eaves."
Rain running off the eaves fell on a strip of ground around the house. Old English called that strip yfesdrype, the "eaves-drip." Word Histories says it was later refashioned as "eavesdrop." The strip mattered legally too. An ancient rule barred building within two feet of your boundary, so roof water would not harm a neighbor's land. The "right of drip" was an easement letting your eaves drip onto theirs anyway.
A person who stood in that strip to listen was an eavesdropper. Etymonline dates the noun to the mid-1400s and defines it as one who stands at walls or windows to overhear what goes on inside.
The earliest eavesdropper on record has a name. Word Histories cites a Nottingham court record dated October 1, 1487. It accuses "Henricus Rowley" of being "a common evysdropper and vagator in noctibus." The Latin phrase means a night-wanderer. Henry Rowley was being presented to the court, not praised.
Eavesdropping was treated as an offence against the peace of a neighborhood. Les Termes de la Ley, an English legal dictionary, defined eavesdroppers in its 1636 edition. They were people who "stand vnder walls or windowes by night or day to heare newes." Their purpose, it said, was to "make strife and debate amongst their Neighbors."
The verb came later, as a back-formation. Etymonline dates it to around 1600, formed from the noun rather than the other way round. Word Histories finds it in George Chapman's comedy Sir Gyles Goosecappe Knight, published in 1606. Momford sends for the key of the gallery and tells Lord Furniwall, "We will be bold to evesdroppe."
Today nobody needs to stand in the rain. The word now covers listening through doors, walls, phone lines and networks. Its picture is still the old one: someone hovering just outside, where the water falls.