I'm sure you can all come up with great ideas, and some of these will seem obvious or passe', but I thought I'd throw them out as a public service.

The Taelon Moon Colony
    This was revealed to viewers in a long-ago episode, but has not been referred to since.  It is an excellent loose end for a roleplaying adventure to tie up.  Players might discover the shipping schedule for supplies or personell traveling to the station and attempt to stow away on a shuttle.
    Any number of weird experiments or strategies could be taking shape on the Moon.  The Taelons could be preparing a military base to resist the Jaridians or building a fleet of warships (or maybe just one big warship).  They could also be secreting human or animal specimens away to this station so they can experiment on them away from prying eyes.

Psychic Phenomena
    Katja Petrenko has proven that the Psionic power of ESP functions in the world of E:FC.  Could Telepathy or Telekinesis also be possible?  A PC might even be designed with psionic powers, though this would probably require a 5-point Unusual Background.  Otherwise, another psychic could emerge to threaten the Synod and the PCs could be called in to deal with him/her.

The Planet Taelon
    Da'an once presented a holographic image of this planet, which appeared to have a rather thick, purplish atmosphere.  This raises interesting questions: Has the Taelon homeworld been abandoned?  Do the Companions have interstellar colonies?  If so, why was their leader Quo'on present on Earth?  Or perhaps the Companions in the Homeship orbiting Earth are all that remains of their race.  Maybe the Taelon population was never much greater than a few hundred at any point in their history.
    This seems unlikely to me, however.  I think it likely that the Taelons control many different planets in different star systems, and each of these is a potential adventure seed.  PCs with scientific backgrounds could be taken to these worlds on a mission of good will, or they could simply steal or stow away on a Companion starship.

The European Companion(s)
    Siobahn Beckett has referred to at least one Taelon with European jurisdiction.  There may be one for the British Isles and one for the mainland, or an entire host of them.  None are named or described, so these Companions have unlimited potential for a creative GM.  (Could there be a South American Companion?)

The Jaridians
    Though it seems unlikely that another Jaridian probe would have been sent to Earth, anything is possible.  Players could hunt down another such craft.  Or these creatures could attain interdimensional flight earlier than was anticipated and arrive in the Solar System in force.
    Are the Taelons telling the truth about the Jaridians' disposition?  Could they be a peace-loving race, and the Companions the aggressors?  Or maybe they are aggressive but were provoked to it by some Companion atrocity in the distant past.