Can anyone think of a game mechanic that would allow you to help out an underdog covertly?
Not really a game mechanic, but you could technically build a wall of spaceships around the underdog; as long as you were not allied or at war with the stronger party, they wouldn't be able to attack the weaker party. Of course, the reverse also holds. You could also theoretically hire some of the other AI factions to assist the underdog, and you could attempt to bribe the strong party to make peace with the weak, but those tended not to work out very well.
I would, of course, like the ability to have some form or another of covert operations that was more reasonably priced and less easily countered than GalCivII's espionage system. Especially something that allowed me to designate ships as privateers (hunt targets I designate as valid under no specific nation's flag; chance that they might turn true pirate) or mercenaries (ships/fleets which other factions can hire to fight under their flag - for a fee, and for a limited time period).
Having the ability to fly ships under false flags would also be neat, especially if you didn't necessarily need the nation's permission to do so (although if you didn't have their permission, I'd want there to be some kind of backlash associated with doing so). It would also mean there was a reason to purchase fleet elements off of various nations, since the more convincing covert operations would use ships in the style of the faction you're trying to frame for the attacks. Might be interesting as an alternative way of starting a war between two factions, too - or even of starting a war between your own faction and another faction and blaming the other faction's "aggression", if there's a reason in-game to avoid being seen as the aggressor.