What the heck... I will post what I have now, maybe some additional/alternative thoughts later. At least you will get some ideas I hope.
DISEASES:
Ancient - Lepracy: "a unit was killed by Lepracy". You will lose just one single (random) unit.
Classical - Arthritis: 35% of your workers in one random city work 50% slower because of Arthritis for 20 turns.
Medieval - Plague: Maybe worst catastrophe in human history. Each city loses 10% of population for each element if it has: open borders, road, trading post, harbor, traderoutes to/from different locations. For example if city has 1 trade route to city-state and another trade route from another civ, plus road to neighbor city, it loses 30% of the population. Max 60% of the population can be killed. Units count as population if garrisoned inside of city.
If the city has a castle, it suffers 10% less (i.e. 10% can be deducted from the accumulated percentage).
Although, if it is possible to actually get infected from religion spreading, or trading over some time period, even better.
Renaissance - Pox: All your coastal cities suffer from pox. You cannot build a settler and the city does not grow for 10 turns. If city has a harbor, 1 population is killed.
If city has a university, the time is reduced to 5 turns.
If civ does not have any coastal city, then it is the capitol.
Industrial - tuberculosis: One random city is hit bad, and another city some.
30% less production for 15 turns for bad.
15% less procution for 10 turns for some.
If at war, units fight 25% less effectively for 10 turns.
With biology you suffer 5 turns less.
Modern - Cancer/HIV: Only civs with happiness 4 or more are affected.
If a city has population of 5 or more, it loses 1 faith and 10% culture per turn for 15 turns.
If city has a hotel, then it suffers 5 extra turns.
Atomic - Animal Flu: Each city with population 10 or more, or with airport, will lose 2 population. Tourism in such city is only 50% for 20 turns.
If you have ecology, you suffer 5 turns less. If you have penicillin, you suffer also 5 turns less. If you have a medical lab, you only lose 1 population.
Information - Jungle Fever: 2 random city-states are completely destroyed (maybe the ones next to a jungle tile). (This also therefore affects UN.)
Each civ that had any contact (pledge to protect, or trade etc) will lose 10% of tourism permanently.
Also 10% of population of the city in contact dies.
If civ has nanotech, then only 5% dies and toursim is affected.