The setting of my story takes place during a future Ice Age, where the equator and parts of the southern hemisphere are a safe haven for humans. Which countries or regions would most likely be the 'superpower' in this world?
Umbrella_Programmer's suggestion of Brazil is a great start. It's a large country, rich with resources, with most of the country below the equator and a bit above.
But since the question asked for countries or regions, not just one...
I will add:
- Island states of Southeast Asia
It appears your ice age related changes are not as strong in the southern hemisphere and Australia is not terribly far south of the equator. Its northern edges are about the same distance from the equator as southern Mexico. Its current population centers are about the same distance from the equator as southern California and the American southwest, or Morocco.
Australia has a lot of land and farms and room to take in refugees from all over. If their focus is on refugees with scientific knowledge or other useful skills, Australia's prominence in the world will rise.
Indonesia is smack on the equator and quite close to other small but resourceful island governments: Singapore, The Philippines, and Malaysia, plus some others. It's also close to the peninsula with Vietnam, Cambodia, Laos, and Thailand.
I predict a Southeast Asian coalition of island governments, which may also include some nearby mainland ones. While some of southern China is close and will be okay in the Ice Age, the country as a whole will be reeling from the effects and focusing on the needs of its enormous and concentrated population. This is the time for the rise of small governments that can work together and take advantage of the changes.
Because of the Ice Age, the islands will get bigger. More land for housing/refugees, farming, what have you. The distances between them will shrink. In some cases, there may be land bridges between islands or even to the mainland, depending on how much the oceans retract.
It depends on your world's economy. Think about how the ice age would restrict things like oil production (which happen in Russia, Scandinavia, the US, Canada, and the Middle East). Think about the sudden dearth of usable lumber, of commodities like grains (wheat, corn, rice, etc). Think about cotton. All of these things require hospitable and temperate climates to cultivate.
So, a country like Brazil, which would have access to some of the last remaining unfrozen forests, huge swaths of land for agriculture, and a large population (and therefore a large army) to protect it all would probably become a global superpower.
An ice age would be coming on for several thousand years, so humanity would have lot of time to adapt to it, or fight over how to adapt to it. So, either you'd have a peaceful cooperation between most nations in order to ensure survival of as many people as possible, or you'd have the military capable nations overrun those who are geographically better situated and resettle their population there. Maybe even humanity would have the time to emigrate to the stars - or burns enough fossile fuels to counter the cooling.
What you seem to think about, the rise to "power" of third world countries, as depicted in "Day after tomorrow", wont happen, since the change in climate is way slower.