A nuisance today is a buzzing fly or a car alarm. When the word arrived in English, it meant actual damage.
English took it around 1400 from Anglo-French nusaunce, "harm, wrong, damage." Behind that sits the Old French verb nuire, "to harm," and behind that the Latin nocere, "to hurt." The first English sense was "injury, hurt, harm," with nothing playful about it.
The word then spent three centuries slowly losing its teeth. By the 1660s it covered anything offensive to a community, such as foul smells, pests and eyesores. In the 1690s it began to be applied to people. By 1831 it could mean something merely disagreeable to one person, the sense most of us use today. Merriam-Webster describes this as a weakened meaning that became dominant in the early 1800s.
Merriam-Webster also notes a likely nudge along the way. In Anglo-French, the forms of this word overlapped with noier, "to annoy, vex," and that overlap seems to have colored how it was used.
The older, stronger sense never fully died. It survives in law, where a nuisance is something that unreasonably interferes with other people's rights. A public nuisance interferes with the health, safety or comfort of a community and can be treated as a crime. A private nuisance interferes with someone's use of their own property and can support a lawsuit.
One branch of that legal family is the "attractive nuisance": a hazard on someone's property that may lure children who cannot judge the danger. Merriam-Webster traces the doctrine to an 1873 United States Supreme Court case about a child hurt on a railroad turntable.
The Latin root has a grim ancestry. Etymonline links nocere to a Proto-Indo-European root meaning "death," the same root behind Latin nex, "violent death," and Greek nekros, "corpse." So a word now used for a mild irritation sits in the same family as necropolis.
That family also explains a neat contrast in English. Noxious means harmful, while innocuous, from Latin innocuus, means harmless. Nuisance sits between them, a word that started near the noxious end and drifted most of the way toward the other.