Due to multiple factors, expect to potentially pay more when booking in January. For low rates, April can be a great choice for travel to Menorca. Flights to Menorca from New York in April are about $862 on average during April, but can be found for as low as $337.
Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from New York to Menorca as you normally would.