Must haves
Sumer: A long lasting, truly ancient civ, invented the wheel
Babylon or Assyria: we can't include three civs from the same region, but one of these definitely should be included in addition to Sumer. This is the area (in addition to Indus, Egypt, and China) where civilization was born.
Carthage/Phoenecians: again an ancient and truly important civ. could be considered an african civ because of Carthage
Turks or Ottomans: a must have, long history with impact, big empire
These all have a long history and were one of the most influential/powerful civilizations of their times.
Others that deserve consideration:
Moors as an african civilization with IMO more impact than for example the Zulu
Ethiopia for similar reasons
Khmer: longish history and Angor wat
Vietnamese, Thai, Koreans
Maybe one of these for geographic balance, though I don't think think they necessarily had enough impact to be included
Celts and Vikings/Scandinavians because they are just so fun to play.
Mycenaean or Minoan civs could also be considered at least in preference to those mentioned below.
Now for those that IMO don't "deserve" to be included
Canada: A recent country and not even one of the most influential ones in modern times. I think civ already puts too much emphasis on modern times
and european based civilizations (Sumer isn't included but Spain is).
Besides it's a separate country and not a separate civilization from USA or Britain.
Australia for similar reasons.
Israel/Hebrew: As a people they have had a huge impact on other civilizations starting with rome. Judaism as a religion already represents this impact in the game and is included when for example sikhism with more adherents is not. But they have never had a lasting empire and have ruled only over a very small area for limited periods of time. The truly remarkable aspect in the people is that they have survived as a small and often persecuted diaspora for a very long time, it is not their success in empire building.
Dutch: too short history, too little impact, too modern and too european
I would sooner drop Spanish and French and maybe even the English than add any other relatively recent european civilizations. The history of civilization is about 6000-8000 years and western Rome fell only 1500 yrs ago.
Byzantine: They called themselves romans, not byzantines. Rome is already included.
Polynesians and other civilizations with little or no impact shouldn't even be considered. Parthians are more fun.