The answer to this question is relatively simple, but needs a historical context in order to be fully appreciated. The short answer is this: October is dedicated as the Month of the Rosary because we celebrate the memorial of Our Lady of the Rosary each year on October 7.
To understand this feast, we need to go back in time. In the 16th century, the Islamic Ottoman Empire presented a serious military threat to western Europe and sent a fleet of ships to attack Christian defenses in southern Europe. Pope Pius V, recognized the grave danger and organized a fleet, called the Holy League, to confront the navy of the Ottoman Turks.
On October 7, 1571, the two navies engaged in a pivotal battle that would determine who controlled maritime traffic in the Mediterranean, Adriatic, and Aegean seas. The fate of western Europe depended upon the success of Christians in this navy battle, which involved more than 400 warships. (This was the largest naval battle in Western history for centuries). Pius V, knew he needed more than just military strength to defend Christian Europe, so he asked that all the faithful pray the rosary, requesting the intercession of the Blessed Mother.
At the end of the Battle of Lepanto, the Holy League was victorious and the maritime expansion of the Ottoman Empire was permanently prevented. The next year, Pius V, established a Feast on October 7 in honor of the Blessed Mother, originally called Our Lady of Victory. After a few centuries, the name was changed to Our Lady of the Rosary, to more clearly recognize that prayer was the greatest power at work that day on the seas.