In most neo-Pagan and Wiccan traditions, the names given to the various moon cycles are based upon a couple of different sources. Some come to us from the Native American tribes of North America, and others are rooted in Celtic and western European mythology. In the Native American tribes, the moon cycles were used to keep track of seasons, and thus designated different agricultural markers. If you live in the southern hemisphere, however, your seasons are directly the opposite of those in the northern hemisphere, and so it wouldn't make any sense for you to celebrate an September harvest moon if September is when you do your planting, rather than your harvesting.
Because of this, people who live in the southern hemisphere would have to calculate their moon names based upon seasons. A lunar month is only 29 days long, so the full moon falls different days each year.
If you want to use the common neo-Pagan names for the moon phases, you can calculate what they'll be based upon the timing of the equinoxes and solstices. The autumn equinox is in March, in the southern hemisphere, so the moon nearest that would be the Harvest Moon. The next one, which would fall in April, would be the Blood Moon, followed by the Mourning Moon. The next month would be June, which is the time of the Winter Solstice in the southern hemisphere, and corresponds to the Long Nights Moon, and so forth.
Moon naming varies from one region to the next, however, so if you're one of those folks who lives below the equator, you may want to look at some of the naturally occurring biological cycles in your area. Another option would be to look at some of the local cultures -- perhaps the people indigenous to your region had their own names for moon phases, which would make far more sense than using the names of people who lived on the opposite side of the world.
There's also some great information about the moon and how it's seen in the Southern Hemisphere at Southern Sky Watch.