Yemen declared a state of emergency on Thursday for Socotra, which lies between southern Yemen and the Horn of Africa and is renowned for its unique animal and plant life.
Largely untouched by Yemen's three-year-old war, it is under the control of the internationally-recognised government whose president, Abdu Rabbu Mansour Hadi, is in exile in Saudi Arabia.
The storm flooded Socotra's villages and capsized boats, leaving much of the island without access to communications.
Mekunu was expected to weaken to a tropical storm before reaching southeastern Saudi Arabia on Saturday, according to the kingdom's meteorological authority.
Yemen is already grappling with one of the world's worst humanitarian crises. The war has killed more than 10,000 people, displaced three million others, triggered a cholera outbreak and pushed the
impoverished country to the verge of starvation, according to the United Nations.