Rather than carve a few facts into a stone monument placed near a grave, carve detailed stories into wood, turn the wood to stone and surround the body with it.
The idea is to encourage people to think in terms of geological time when they consider their deaths and what stories they hope might last. The promise is that perhaps some meaningful part of them could realistically last millions of years. Would they tell their own story or perhaps something larger? How could they make themselves understood long after English has disappeared? What is the most important thing they could say to the distant future? What should they really care about when they think about their end?
Stories, images, hieroglyphs can be digitized and formatted to fit onto casket sized boards and carved in relief using cnc routers. There is little practical limit to how much information can now be carved into a few wood planks. It would not be too expensive to carve long stories into many boards and build nesting, Russian doll style caskets, urns or ossuaries if someone had a lot to say. Alternatively people could be buried with wood tablets that are on their way to turning into stone.
The boards can be pretreated in mineralizing baths before being assembled into boxes for the burial, or the treatment might be held until the cadaver is inside, thus perhaps fossilizing the body as well. Deposition of silica can be induced to start very quickly. It takes on microscopic detail of the cell structure and the surface of the wood. For the wood to truly petrify into the form we are familiar with takes many thousands of years but a strong substrate is established quickly. It is even possible to establish a very hard ceramic shell in the shape of the wood if it is repeatedly soaked in minerals then heated.
Caskets of those hoping for petrifaction would have to be buried in carefully considered locations with concern given to whether the process is likely to continue long enough without disruption. Should there be fossilizing cemeteries? What are the risks of erosion? Is there dissolved silica in the ground water? If there is lots of dissolved oxygen near the wood then rot will set in quickly and the wood may collapse before much more mineralization can occur. One thought is to bury the casket in volcanic ash, the stuff that fossilized the unfortunate residents of Pompeii.
The process of petrifaction is well studied. There is even a patent on one recipe. See: http://petrifiedwoodmuseum.org/pdf/permineralization.pdf