Opening up a file in a recent game and reexamining a Civil War I staged; and then using Cheat Mode to restage it for purposes of an experiment, the following are my observations.
1. (As mentioned above). There has to be an open slot for that civilization. I also think that the "Don't Restart Eliminated Civilizations" has to be toggled off if you started with the full complement of seven civilizations.
2. The opposing civilization must have less than 1000 gold, otherwise it will move its capital to another site.
3. When I staged this Civil War, the civilization that suffered the Civil War was the highest ranked of the AI civilizations; I was in fact the highest ranked of all the civilizations -- I originally staged the civil war as an exercise in perversity (I was having one of those days).
4. At the time, the opposing civilization had 10 cities; the four cities nearest to the capital that fell remained loyal; the five furthest away defected; a half & half breakdown
Of the defecting cities, the nearest two of these cities were six squares from the capital that fell; the furtherest of the loyal cities was only five squares away.
5. I'm not sure if happiness or type of government has any specific effect on the chances of civil war; in the above case, the government was a democracy; and generally speaking, examining their cities prior to the Civil War revealed that most did not have unhappy citizens, and only their smallest city was suffering from Civil Disorder. I'm inclined to think that the inability to move the capital (i.e., the civilization has less than 1000 gold) is more decisive than government type and unhappiness.
