The best time to visit Arequipa is through the months of June through September, during the dry season.  That said, Arequipa averages over 300 days of sunshine every year so visiting during the so-called rainy season probably won't spoil your trip.
Each year on August 15th there is a city-wide party celebrating the day Garcí Manuel de Carbajal settled the city in 1540. On this day and those surrounding it the city is packed and last-minute accommodations are hard to come by.  It is a great time to see Arequipa if you want to mix with lots of locals and see how they kick-back, however if you are planning on being in Arequipa in mid-August be sure to arrange accommodations well in advance.