In the northern hemisphere, early Native American tribes called this the Worm Moon because this was the time of year when the ground would begin to soften and the earthworms would reappear.
In the southern hemisphere, we might call it The Full Fruit Moon due to the wide variety of fruits harvested throughout Australia and New Zealand in March. Different climates dictate the type of fruit most associated with the regions of these two countries. So this could be easily called the Full Grape Moon, Full Apple Moon, Full Melon Moon and so on. Some folks tried to make this simple and lumped everything into one basket and simply called it the Fruit Moon.