Babylon - A grand Mesopotamian Empire with a capital being a city of wonders, the largest city in the world and important centre long after the fall of both Achaemenid Persia and Alexander's Empire. Methinks with a bit of work, it could be made different from Sumer. (let's replace the Babylonian monument could be replaced with the law stele). About leaders, Hammurabi, Nabopolassar and Nabuchadnezzar II would all make great leaders.
Portugal - a small Iberian country successfully managed to overcome Ottoman high taxation of European spice traders, dominate the naval exploration and form a great trade Empire, and also the very first colonial empire in the world. Manuel I for me, with the unique and beautiful Manueline architecture incorporated into Portugal as his agenda or leader bonus (and/or in the Portuguese palace design) would be great. João II could also do.
Maya - I'll admit here that I have little to no knowledge of Mayan leaders, so anyone good will do.
Byzantium - survived almost 1000 years longer than the Western half of the Roman Empire, successfully influencing Eastern Europe by spreading Orthodox faith to Balkans and Russia. A high medieval Byzantium would be great this time. I would like to see Basil II Bulgaroktonos, Alexios I Komnenos or Manuel I Komnenos.
Ethiopia - one of the most ancient Empires to exist, a bastion of Christian faith in Africa, the only nation in Africa to never get colonised, successfully humiliating the Italians at Adwa. Menelik II would make a great leader. Also, I want that string instrument from the Ethiopian war theme back.
Morocco/Moors - A rather empty part of the world could be very well filled by them. I also love their architecture (well, and I also seem to be probably the only person to seriously like their Civ V themes

) I have little knowledge in the leaders of Morocco/al-Andalus, so I suppose anyone good would do. Even Ahmad al-Mansur again, if they make him more militaristic (I mean, he conquered Songhai just for those gold mines...)
Mughal Empire - Once again, I adore the great Mughal architecture. The Mughals were incredibbly rich. Also probably the closest to unify India since the Maurya Empire. Akbar the Great is my preferred leader choice. Din-i-Ilahi offers itself for his agenda/ability immediately.
some Native American tribe (not excluding
Navajo though) - I aknowledge the emptiness of this area and I think adding some tribe would be nice. Unfortunately my knowledge/interest in Native Americans is low, so I don't have any specific wish.
There are many other great choices to pick tho... Bohemia, Austria, Italy, some Balkan Empire other than Byzantium (Serbia/Bulgaria), Assyria, Armenia...
Should I also pick an alternate leader, I would go either with Thutmose III for Egypt or a Ming/Qing/Song/Tang Emperor for China (preferably Yongle of Ming or Kangxi of Qing)