What I'd also like to see is sequential or otherwise related random events - say, you get the "jilted noble bride" random event, that gives you a % chance that you get a later event where one of the bride's family members murders the groom as an act of revenge, which can set off a string of feud events. If you attempt to make peace, you could improve relations with another civ and end the feud - but the other civ could be milking it to worsen relations, counteracting your efforts (making peace would make the other civ happy with you, but your noble family wouldn't appreciate your meddling in their affairs, giving you an Unhappiness penalty; therefore, allowing it to continue or actively encouraging it would give you Happiness, since your nobles like it when you support them).
This would allow you to affect other empires as well as your own, depending on your decision; if they keep trying to make peace, they're going to keep renewing their Unhappiness penalty, and if you're stoking up the fire on your end it could eventually result in serious trouble for the opposing civ, such as a building (funded by their nobles) being destroyed, or even a rebel unit (common people fed up with the feuding or incited by your agitators) appearing outside an enemy city. It wouldn't dominate diplomatic relations - a noble feud wouldn't outweigh good trade relations or a Declaration of Friendship - but it could be a way for you to sabotage the enemy before espionage proper appears, or alongside existing espionage efforts.
Certain sequential events, if properly managed over a long period of time, could even be used to weaken a larger, more powerful civ, perhaps even by souring relations with a third party until they go to war, allowing you to jump in and take over some of their land without getting horribly curb-stomped.
Of course, sequential events don't necessarily have to affect other empires as well. For instance, if you get the "Collapsed Mine" event and spend the required money to rebuild it, you could later get an event where civil engineers have studied the mine's collapse and the rebuilding effort, and come up with a plan to increase mine safety across the board, allowing miners to dig deeper and further without risking disaster. If you spend the money (depending on the total number of mines in your empire), mine yield could be increased by +1

everywhere, or you could choose to just improve mine safety without digging deeper, giving +0.2

per mine, or you could choose to sell the plans to mining companies (instant lump sum of gold but +0.1

per mine), or make them publicly available (+0.1

per mine for all known civs and/or +1 diplomatic relations with every known civ).
Other sequential events could be a volcanic eruption, followed some time later by geological studies in an attempt to predict the next eruption. Dropping some money to fund ~3 such studies could give a prediction with accuracy within a handful of turns, which then allows preparations to be made when the next eruption is due, which would reduce the damage from the eruption and possibly increase happiness as well.
I may be getting a bit sidetracked, lost in the details, but that's just how my mind works. In any case, my point is this: In addition to "just" random events, I'd very much like to see random events that trigger (or give a % chance to trigger) other events further down the line, that build upon the (decision made in) the earlier event. I fully understand that any official implementation of this would be quite limited (as I recall, BtS did have this ability, but I only encountered one event (something to do with bandit activity and a lack of food) that had followup events, and the quests of course), but having the framework in place, especially if specific decisions can be tracked as variables, would be a great boon to modders.