Monsoon began as a word for a time, not a weather event. The Arabic mawsim means "season," specifically a suitable season. Etymonline traces it to a verb meaning "to mark," so a mawsim is a marked-out stretch of the year.
The word became a name for wind because of how sailing worked in the Indian Ocean. Etymonline describes the monsoon as an alternating trade wind there, and says the season of the southwest wind, roughly April to October, was the one that favored voyages to the East Indies. For sailors, the season and the wind were closely tied.
Portuguese sailors picked the word up in the Indian Ocean and rendered it monção. Etymonline says it came to mean anything that recurred yearly, and then the southwest monsoon season that favored voyages to the East Indies. English took the word in the 1580s. Etymonline lists Dutch monssoen and Portuguese monçao as the immediate sources. Wikipedia says English took it from Portuguese, partly via Dutch, and calls the etymology "not wholly certain," so the exact route into English is less settled than the Arabic root.
The meaning then drifted. In India the summer monsoon is much stronger than the winter one, so people called it simply "the monsoon." Because that wind brings heavy rain, the word came to mean the rainy season itself. Etymonline dates the sense of a heavy rainfall to 1747.
The American Heritage Dictionary agrees on the Arabic source, mawsim, "season," and derives it from wasama, "to mark." That root gives the Arabic root wsm.
What survives is simple. A word that meant "the marked time" passed from Arabic to Portuguese in the Indian Ocean and reached English as the name of a wind and then a flood.