Let's assume this happened tomorrow, and it took another 500 million for life to evolve to what we would consider intelligent. Would anything of our civilisation be left? What would be the best way to leave a record that we existed?
Just keep going business as usual. Given that we have plenty of trilobite fossils from the Cambrian, might even expect future paleontologists to find some fossilized skulls.
The only problem with this is that we might not have long enough for multi-cellular life to recover. While the red giant phase is a long way off, some scientists think that the carbon cycle could shut down as little as 500 million years from now, either due to oceanic evaporation or due to a failure in the weathering process.
I think we might be earth's last chance at an advanced civilization. We've depleted all of the readily available energy sources necessary for an emerging civilization (easily-accessible fossil fuels). It will take at least 500 million years for them to be replenished. And in one billion years, the increased luminosity from the sun will wipe out most advanced life due to runaway greenhouse effects. So there's a pretty narrow window in-between.
We could launch 10,000 time capsules into space on trajectories that bring one of them back to Earth every 50,000 years or so. They would have to be designed in a way that makes re-entry spectacularly visible so they would be noticed and it would be possible to track them down. Not perfect, but at least there is some chance that one would arrive while an intelligent species is thriving.
