"Posterra" rolls nicely off the tongue, but reminds one of the phrase "post mortem" with respect to the Earth. The name gives me a sense of urgency about the organization's purpose. Why not a name that means "beyond Earth" rather than "after Earth"?
Plugging in "beyond" at The Latin-English Dictionary or a search at the the Latin Dictionary and Grammar Aid yields a few possibilities--
From another perspective, perhaps "Posterra" invoking the association of 'post mortem' at some low subconscious level is not a completely undesirable result. Humanity can not forever remain in a position of superiority and mastery of planet Earth. There are limits to the habitability of Terra, if none other than the expansion of the sun into a red giant in 5 or so billion years.
But there are other threats with likely occurrences which are much nearer in time--coalescing binary neutron stars in our neighborhood (timescales of < 1 billion years for significant impacts on Earth's biosphere), migration of the ecosphere away from Earth's orbit (<~ 1 billion years), impacts of large comets or asteroids (10s to 100s of millions of years or less)...these ultimately threaten humanity if we are to survive for so long.
The astronomical threats to humanity are not even the most pressing--exhaustion or depletion of concentrated naturally occurring resources, contamination or poisoning of the biosphere, disruption of the food chain, unbalancing of the O2/CO2 cycle, threats (both direct and indirect) from our own species, overpopulation and collapse of the flora/fauna balance, and many others dangers. Listing those possibilities would show some as far fetched to some minds, but given our brief tenure as an aware species on Earth, there are bound to be some risks that we aren't even aware of yet.
The name "Posterra" serves to remind us that if we do not find a way to leave home and nurture Earthlife in a new cradle, then we run the risk of having aliens some 3 or 4 billion years in the future landing in their space ships to dig through the fossil record, conducting their own post mortem of the fate of life on Earth. Posterra is not just a game; it is about life and death of a species, and of a planet, and of the life that arose on that planet. Over the long term, the stakes are absolute and extreme. This entity needs a name that is exactly that serious. That sense of urgency needs to be built in to the organization.
