Hello Everyone.
I think this is an excellent question, both for fun and for gameplay. I think the best way to help the OP (if he's still reading) wouldn't be to list out the civs
the best thing that would help the OP would be Munch's method (listing out by continent) because of course he's going to have to narrow the list based on gameplay concerns. I feel like just a listing of the countries that really should not be excluded if at all possible would be the best. I also think that Iranon's approach is best: pick one of similar cultures, no Rome+Greece, just one.
My own list:
1. China. Enormously long history as a major power, from ancient to modern times, huge cultural contributions, dominance over a ridiculously large area.
2. Romans. Greeks were great and all, but the Romans rose to power (semi)independent of the Greeks and had their own political structures. It was mostly the fact that the Romans liked their gods, art, and sense of history that we still think of the Greeks as being enormously important. The Romans formed the foundation of western civilization. If they hadn't broken apart they'd be #1.
3. England. Enormous empire, industrial revolution, hugely influential culturally, about 400 years as a major world power, and with the time bias to the game (how the years slow down) that's almost as good as the Romans.
4. Egypt. Another very long history, though they weren't so important after ancient times. Mostly this is a vote for them over the Babylonians/Sumerians, who had shorter runs.
5. America. Current most important nation, dominated most of a century, globally important culturally and economically.
6. India. Spent a long time as a colony, and hasn't been strongly united very long, but they're very important today, and their culture is of enormous importance to about a billion people.
7. Arabs. Enormously important scientifically, still a fairly coherent "culture" across a very large area, second-most popular religion in the world. Historical powerhouse for a while as well. This pick subsumes the Turks, as well.
8. Spain. Gigantic empire, most important nation in Europe for a few centuries, culturally of great importance, fell off in the last few centuries.
9. Mongols. Significant mostly for the destruction they rained down rather than their cultural achievements. They allowed people to keep their religions and cultures, however, and so their long-term impact on that front is questionable.
10. Poland. Just kidding. Seriously, I tried to fill this 10-spot, but most of the civs left either didn't rule long enough/"hard" enough (Dutch, Portugese, Native American, etc.), are fairly redundant with the ones above (Byzantines, Turks, Celts), or had limited cultural influence (Vikings, Zulu, etc.). If pressed, I would say Japan is the next-most deserving.
A quick comment on England... even though I still rank them high, the argument that they set up their colonies to succeed is weak at best. The real reason those colonies were not economically devastated and were usually allowed to leave without much of a fight (at least Canada/Australia) was because they were colonized by white people, English people in particular. English colonization may have been less oppressive than French or Spanish, but it was still very damaging to the countries that were ruled by a small white elite (Nigeria, Ghana, etc.).
or that France= Poland , because of the father in law of Louis XIV had become King of Poland and the crown of Poland had fell on the Bourbon family ( I'm pretty sure you know that story better than me 
).... ).
Or Iron working for that matter.
