In my development class I have my students read Paul Collier's book The Bottom Billion, where Collier's hypothesis is that revolutions/civil wars/coups are the result of low gdp per capita (being poor) and low gdp growth (bad economic times). He has academic works to back up this hypothesis.
The economist magazine has come out with its own index for the likelihood of a revolution the shoe throwers index. I'm not sure they have tested it yet against data, but their method is "weighting of 35% for the share of the population that is under 25; 15% for the number of years the government has been in power; 15% for both corruption and lack of democracy as measured by existing indices; 10% for GDP per person; 5% for an index of censorship and 5%"
Sounds like a good start. In the comments people suggest % of the population that is college educated and internet usage. I'm sure will see some academic papers on the topic soon.
h/t Texas in Africa