Eavesdropping originally came from Anglo-Saxon laws against building too close to the border of your land, so the rain running off your roof would not run on to your neighbor's property. The eavesdrip is the width of ground around a house or building which receives the rain water dropping from the eaves.
The primary function of the eaves is to throw rain water off the walls, prevent erosion of the footings, and reduce splatter on the wall from rain as it hits the ground.
Eavesdropper became the word for a person who stands within range of the eaves drip in order to listen to what is said inside the house. Now it has evolved to the act of secretly listening to the private conversations or reading online records of others without their consent. The word eaves is both singular and plural.