I seem to have given the ancient civs with few surviving remnants, or limited global expansion, a fairly raw deal - more of these should probably have been in my top few. However:
13. Assyria: I'll try to rectify the modern bias a bit. As an ancient empire, Assyria was one of the most extensive and influential - on reflection it should probably come above Egypt. Today it's probably best-known for a tale dating back to Sumer, the Epic of Gilgamesh; the first known copy was found in the remains of the Royal Library of Nineveh, itself a likely inspiration through legend for the later Library of Alexandria.
14. Byzantium: Byzantium did indeed outlast the Roman empire for centuries; for much of that time it was weak and seen as a relic of the old order, but it was still a bastion of Christianity and the impetus for several Crusades, and its form of Christianity survives today. Possibly its greatest significance comes from its place as a buffer against invaders like the Sejluks and from its eventual collapse, or more specifically the identity of the culture that replaced it.
15. Mongolia: Much of the lasting Mongol legacy can be considered Chinese (as the Mughals can be considered Indian), but while today the majority of traditional Mongol territory is sparsely-populated and of limited geopolitical relevance, the area itself is vast and groups like the Mughals, who traced their origins to earlier Mongols, had substantial effects in their own right.
16. Portugal: Even during the height of its empire, Portugal was never considered a major power on the European stage; in Britain we like to think that we're the small nation punching above its weight on the imperial stage, but Britain is a gigantic island (the 9th largest in the world) on a small continent and England, while waxing and waning, has always had its place as a major power in Europe. That Portugal, a minor former outpost of Moorish Iberia, was the first European power to venture far outside the continent by sea, is a much more astonishing achievement, and was considered so at the time. The caravel, the symbol of the Age of Discovery, is a Portuguese design. Both contemporary and later commentators have given very negative impressions of Portuguese colonialism; without Portugal there might have been no international slave trade, active Portuguese interference is blamed for the collapse of the Kingdom of Kongo and the ultimate cause of civil strife that persists in parts of the DRC, and until recently in Angola, and few 19th Century British observers missed an opportunity to contrast their (and in some cases, like Wallace's, the Dutch) colonial enterprises with the zealotry and barbarism of the Portuguese and Spanish. But without Portugal they probably wouldn't have been in those parts of the world themselves. Portugal's legacy only persists today in Brazil (for the Portuguese themselves in their heyday, an area of less importance than Mozambique and the other African territories), but they set the stage for European colonialism.
17. The Ottomans: As Morocco is designed to represent mainly the 17th and 18th Century state - and, as the game's only Berber representatives, the Moors only by association - the Ottomans are indeed the most important Islamic power after Arabia. As well as being a powerful, wealthy and long-lived power - and a significant player in the early stages of European colonialism, which was inspired in part by a need to break the Ottoman monopoly on trade with the East - its terminal decline was one of the complex of factors that led to a series of wars involving the Western powers, the most significant of which were the Crimean War and World War I.
18. Persia: Persia's place is long overdue, but again hard to gauge. Persia has had many incarnations and is one of the oldest surviving civilizations in the world (as reflected in Civ V, which gives it a medieval UB). In ancient times it was responsible for the destruction of Babylon, and has long shaped regional politics - early in the 19th Century, the Persian defeat of the Maratha prevented what may have become the first fully unified Indian state. Persia's story is of course not over - Iran (the inhabitants' own name for the country since the 2nd Century BC) is now the major power in its region once again. Much of ancient Greek historical development is a consequence of rivalry with Persia, including Alexander's imperial ambition (which began as a campaign to destroy Persia).
19. The Netherlands: A late and like Portugal unexpected arrival on the European colonial scene, but along with Britain one of the major European empires of the 19th Century, and with Portugal a formative influence on immediately post-Sengoku Japan.
20. Ethiopia. The oldest indigenous African kingdom, and an earlier focus for European exploration as a fellow Christian state (at one point identified with the mythical Prester John). Modern Ethiopia is ultimately the successor to the Kingdom of Axum, one of the greatest of the classical-era empires, and a founder member of the United Nations.
21. Carthage: rated as low as they are simply because, aside from the Punic Wars, I know very little about Carthage and its wider impacts. As with Egypt, the territory is no longer culturally connected to the Carthaginian people, making their lasting influence in the region difficult to interpret.
22. The Maya: I've had no neotropical civs yet, so here goes. Very recent excavations suggest that a form of city planning characteristic of Middle America, with pyramids set around a central plaza, may in fact have originated with the Maya rather than the Olmec as once thought. The Maya were the longest-lived civilization in the Americas, and while the continents' isolation and their own restricted area in the Yucatan Peninsula mean that the Maya had very limited global influence, both their monumental architecture and their religion inspired later societies in the region.
23. Denmark (taken here to equate to "Vikings"): if influence was defined by pop culture recognition, the Vikings would be in the top 10 if not the top 5. Their genuine influence was substantial, but geographically limited (the founders of Russia were from what is now Sweden, and while obviously culturally related were not noted for longships and pillaging). Danes explored and colonised a wide area of northern Europe, and through their derivatives the Normans extended their range further still. Given later developments, their most important acquisition was Britain - with the Danelaw established in most of the island, Norse occupation of Ireland, and a secondary Norse (albeit by that point culturally French) invasion of England by the Normans, Britain still has a significant Viking legacy; in English the days of the week are still mostly named for Norse deities (although likely via the earlier Germanic invasions rather than the Norse).
24. The Inca: Short-lived and quarrelsome, like the Aztecs the Inca were destroyed by the Spanish with help from local malcontents and, in the Inca case, the fact that the empire was already weakened - perhaps fatally so even without European intervention - by civil war. All that can really be used to gauge the relevance of the Inca is their huge territorial extent; the culture itself was not dissimilar to any of the other, more localised mountain kingdoms that had preceded it, with even its famous brickwork having been developed by precursor societies. There is certainly no doubt that for scale alone the Inca system of urban planning was unprecedented in the Americas. With almost 50% of the country's population of indigenous descent (including a former president), a resurgence of Andean indigenous pride, and place names largely from Quechua, modern Peru is the closest the Americas have to a successor state of one of the native empires.
25. Poland: Based on not much more than a vague feeling they should be here somewhere - of all the game's civs, this may be the one I know least about (except Polynesia).
26. Morocco: and possibly this one - insofar as it represents the Moors, Morocco deserves a far higher placement, however the Civ V design is for a more modern state whose relevance I don't really know enough to gauge.
27. Japan: Okay, I've resisted it long enough possibly as a reaction against the modern Japanese dominance of internet pop culture (and the electronics industry). The truth is, Japan has a long and proud imperial and military history ... and in terms of its importance to the rest of the world, all of it (except WWII, and an earlier invasion of Korea) pales into insignificance when compared with the global impact of the Sony Walkman in the 1980s. Japan remained a feudal society longer than almost anywhere else in Asia; its language and religion are derived from China, and its modern technology from its contact with America in the 19th Century (having adopted little except guns from earlier European contact).
28. Indonesia: I'm a particular fan of Southeast Asian history, and actively agitated for Indonesia's inclusion in the game. Simply because their territory was so large the historical Indonesian states were significant, but it's not clear that they had much of an influence outside this region. The Spice Islands exported their produce to Europe via India, and were outside the practical limits of the largest pre-Dutch Indonesian empire, the Javanese Majapahit. As a modern territory I like to describe Indonesia as the most important country in the world that no one ever hears about - the fourth largest population, the largest Islamic nation, a major Cold War ally of the West (and regional antagonist for Australia), and a founder member of nearly every Southeast Asian charter organisation. And yet, while everyone's heard of it and if nothing else know about its coffee, in the West it receives remarkably little attention.
29. Venice: The last of the European nations of any particular importance, as both Austria and Sweden had rather short periods of dominance, and those regional powers in a globalised world. It was a major medieval naval power; its unsuccessful efforts, in alliance with the Ottomans, to disrupt Portuguese expansion and protect its monopoly were its last actions of significance.
30. The Celts: This far down the list because they're nearly impossible to define - if taken as a surrogate for the Scottish and Irish, they deserve a somewhat higher placement due to the influence not of these countries but of particular individuals from them and their ideas - David Hume, Adam Smith, television, the telephone. Instead treated as an amalgam of prehistoric and classical northern European tribal cultures, they're notable for artefacts and stone circles, but the precise inability to define anything culturally 'Celtic' (except by a modern definition of the term that refers to largely Norse-influenced cultures) suggests their lasting impact wasn't particularly great.
31. Austria: Time to deal with the remaining Europeans. Austria isn't much more than the largest bit of the Holy Roman Empire outside Germany, but as has been noted it has significant cultural figures to its name (not to mention strudel), including a number of its Hapsburg rulers and, yes, Mozart. Vienna has been a European cultural centre for centuries.
32. Songhai: Another empire given its place by virtue of territorial extent, with economic achievements to its name and, under Askia, a drive to modernise along European lines. Sadly it is also one of those cultures whose legacy was essentially erased by European colonialism.
33. Brazil: I hate to say it, as a longtime proponent of "Brazil isn't a civ", but there isn't really anyone left that deserves a place above it. Brazil did have an empire, and has a much larger (land) area than any of the remaining civs. Just as fusion power is 50 years away and always will be, Brazil is an "emerging power" a decade away from being a world power ... and always will be. It's been emerging for so long now (and is part of a 'BRIC' club all of whose other members are much closer to fulfilling their expected promise) that it's a bit of a joke, and Brazil's serious economic downturn in the global recession is in stark contrast to the growth in India and China; if it ever should have been in Civ, it was in Civ IV when the country was genuinely doing well.
34. Siam: While, as noted, I'm a fan of Southeast Asian history, this in itself tells me that (much as I love the civ in game terms) it doesn't really belong. The struggle to find a unique unit other than 'Naruesan's Elephant' is a bit of a warning sign. Plus, the Siam of the game itself is not the precursor of modern Thailand, but the Sukothai Kingdom the Thais (who emerged in southern Thailand and overthrew the declining Khmer there) conquered. Thailand doesn't figure much in the histories of neighbouring countries (at least in Indochina); its main claim to historical fame is being the only Southeast Asian nation to remain independent during the colonial period.
35. Sweden: An important European power during the Thirty Years War, and not much beyond it - although, as mentioned, it is notable for several people, and in pop culture for Abba and Ikea.
36. Korea: Not an especially large or ever influential society, but it was a sophisticated and developed culture in its own right, which counts for something when most of the remaining civs are tribal.
37. I commend the original poster for a well-balanced approach to the Aztecs who, like the Zulu, made it into Civ based on popular recognition and a need for one representative of that particular geography, and somehow never left. And like the Zulu (who were prioritised over Ethiopia), the Aztecs made it in on celebrity status in place of a better option (the Maya, who weren't popularised until somewhat later). Short-lived, locally despised by their subject tribes, geographically limited, and with little tradition of monumental architecture, the Aztecs are an odd fit for Civilization and always have been.
38. The Iroquois: Political correctness demands North American tribes, and of the available options the Iroquois are the closest to a 'civilization' that we know anything about. They may be something of a European invention; the confederacy system emerged following European contact, but was probably a mix of European concepts of social organisation and traditional tribal practices for negotiating agreements rather than either fully European or fully indigenous; and the Iroquois are known mainly for their roles in European conflicts in the Americas.
39. The Zulu: Though essentially tribal, the Zulu did have organised settlements (if not exactly cities), and they were a major influence on the development of British policy in southern Africa. At this point we're pretty much down to distinguishing between tribal societies based on their charisma, and the Zulu definitely win on that score.
40. The Huns: Yep, Attila has charisma as well. If Civ had some way of representing a barbarian 'nation' that launched a coordinated invasion (something more detailed than Civ IV's stack of barbarian random events) and could be negotiated with, but not played, the Huns would be ideal - Total War did something similar with the Mongols. As a civ they don't fit.
41. Polynesia: Yes, a very widespread people, but also very isolated and not a coherent single culture. Although Hawaii did develop a kingdom, this was following European contact and I'm not aware of indigenous communities with an advanced level of social organisation; as urbanisation is the single key criterion of Civ, a game based around cities, all of these tribal cultures are a poor fit.
42. The Shoshone: the stated intent of the civ is interesting, as a 'nice guy whose land you invade' moral dilemma, but as with the Huns this is something more suited to some kind of non-player challenge mechanic, not a playable civ.
Only 42 - who have I missed?