The Sydney Opera House.
Sorry for giving away the answer right away, for those who wanted to puzzle it out. Having front loaded this writeup by giving away the answer right away, I will now explain why that is important.
Even someone who is not particularly knowledgeable in geography and architecture could spin off the names of dozens of buildings in the Northern Hemisphere. Some are notable for their iconic appearance, such as the Taj Mahal, some for their historical importance, such as the White House, and many, such as the Great Pyramids, for both. There are probably a good two dozen buildings that have entered into popular iconography so that their appearance in a movie would be an automatic shorthand for a geographic location. The Great Wall means "China", the Eiffel Tower means "France".
And yet the most famous example of architecture in the Southern Hemisphere, and in fact the only one that has gained truly iconic status, is an opera house in Sydney that is a rather recent invention, and not deeply rooted in the culture it represents.
This is a natural result of the fact that most of the Earth's land area is in the Northern Hemisphere, and all but one of its classical urban cultures was based there. This is not just a eurocentric belief, since all of China, India and Mesoamerica is also north of the equator. Much or most of Africa is also north of the equator. Other than one exception mentioned below, there was little urbanization in the southern hemisphere previous to a few hundred years ago.
Other than the Sydney Opera House, there are a few other famous structures that might be considered, but they all fall short of the Opera House for various reasons.
- The Petronas Towers are certainly an iconic building, and despite their recent construction, they are recognizable, as well as being an architectural achievement. However, the Petronas Towers, located in Kuala Lumpur, are north of the equator, although barely.
- The moai of Easter Island are certainly iconic structures, but they are not, by most standards, a "building".
- Christ the Redeemer, a statue located in Rio de Janiero, Brazil, is also a famous structure, but again, is not a building.
- Amundsen-Scott South Pole Station certainly qualifies as "in the Southern Hemisphere", and the fact that there is a station there is probably something that the average person is aware of. However, the specific appearance of the station is not iconic, or even well-known.
- The Machu Picchu complex, the most famous site of Inca civilization, is in my mind the best qualified runner-up in this category. It is well known, has much historical importance, and its appearance would probably be fairly widely recognized. However, I still think its fame is probably slightly less than that of the Sydney Opera House.