This has been mentioned before - maybe divide Espionage into 2 or 3 catagories for each race: Espionage, Destabilization, and Counter-Intelligence.
Espionage would require a continuous stream of money to maintain your level, or at least a minimum amount per turn. In turn, it would let you gather information about their ships, planets, and starbases, occasionally steal technology from them, and eventualyl give you a massive diplomatic against them. If you built up the level to Advanced (or higher) on every surviving race and kept it that way for 6 months (24-26 weeks), you could win an Espionage Victory.
Destabilization would function similarly. Each race would have a slider you could set each turn. Activiating Destabilization against a given race would cause that race to slowly lose morale points up to a certain point for each level you had in Destabilization (ie -5 morale maximum cap at Low and -25 at High). At the higher levels, it would occasionally give you random events such as a freighter crashing near one of your planets and you recovering all of its cargo, a Planetary Improvement blown up on their home planet, etc. At the very highest level (and it would be set up so it would only work if you also had a certain level of Espionage against that race, so you couldn't destabilize a given planet without having at least a High level of Espionage against the planet's race) you would be able to pick and choose your targets in that race for a cetain amount of money that would be determined by their level of Counter-Intelligence and Espionage against you; options could include sabotaging starships, crashing ships in orbit, deactivating, capturing, or blowing up starbases, and sabotaging planetary improvements (poison a farm, so it stops producing food for 10 turns and kills 2 billion automatically, or scrambling a factory's automated computer system so it produces faulty starships, etc.). If the other race has a High or Very High level of Counter-Intelligence, there is a random, large chance they will discover who destabilized them and teact appropriately (ie war).
Counter-Intelligence would (finally, this is the only thing that really needs to be added) let you counter all of this. By setting your own level of Counter-Intelligence (its expense would vary with how advanced your civilization was and how many races were spying on you), you can shut down Espionage and Destabilization. It would cost about the same as the two others combined, with an added factor of 5% for every level of Espionage and Desbailization a race already had against you.

Having your grubby fingers in every pie up to the point where you are the *real* power behind pretty much any scene I guess

wow, you guys have more ideas than I thought

Recruiting citizens would probably mean those people deciding to move to another system owned by the destabilizing party which means more pop hence more tax. Sounds good.

I'd just like to say that if the espionage was to be given an active role (actually doing something instead of simple info gathering) you need strong counter-espionage options as well. Otherwise espionage would be far too powerfull.

That said, your spies could incite a revolt or just lower the happyness on a planet so that more people would join you in the informational war (invasion).

On top of my head:

- Reduce morale of a target planet. This will only be helpful if low morale will actually hurt a planet. As it is, I've never seen planets go into revolt so the only hit is on pop growth and taxing. I could be missing something, though.

- Cause a planet to change a percentage of the influence to your kind instead of it's native kind (represents it becoming easier to defect to your side). This is a replacement for snatching a planet. Now, you need to be close enough to get it in your sphere of influence. It will only make it a little easier, it won't flip on its own.

- Destroy or block research. Target empire loses reseach points or won't generate them for a limited number of turns.

- Steal tech. Obiquitous aspect of the espionage game. I find it very frustating to lose tech to others this way so I'd love some form of counter-espionage.

- Reveal the position of Galactic Wonder/Achievements or inform when they are being built. Reveal planet with most research/manufacturing/population. This should make it easier to target planets then having to look through all of them.

- If caught, your standing should drop with all races, not just the target. After all, if you spy on one of them, what's keeping you from spying on all of them? There could be a modifier: spying on an enemy at war should not be as 'costly' as spying on an ally or good friend should be. The type of espionage could also be of importance. Plain information should not be punished or very lightly, subversion should be penalised heavily. I'm not sure of this, it might make espionage too costly.