Imagine that you're part of a civilization that is bound to disappear fairly soon, due to (pick one) environmental upheaval, resistant parasites, coordination failure in a nuclear Mexican standoff, or being outcompeted by its technological/memetic offspring.
Yet, like the proverbial dog, you would like to leave a mark of your presence in the universe. Something that better expresses the sum total of your aesthetic, philosophical and scientific achievements than a floating continent of plastic detritus in the middle of the ocean.
Sure, you can put little plaques on space probes, or join any of the Long Now Foundation's art fair projects. But these only work provided that your potential reader's mind and culture functions sufficiently like yours (or is so vastly superior to you that, even with so little data, inferring the evolutionary and psychological prerequisites to decoding your message - and to even recognizing it as a message - is an easy task).
A trending way to encode information, that still carries a little of its former scifi glamour, is DNA. It's robust, replicable and economical, space-wise. And if you're an advanced civilization wandering Earth, sooner or later, you are going to recognize that DNA is (or perhaps used to be) something important that you should want to understand. Stumble upon a book, it's not obvious that all these black scratches are any more significant than a metazebra's spots; however, break down these wood fibers and you eventually get to a universal information-storing engine. Hopefully, life itself is recognizable enough as a phenomenon that they won't miss it entirely.
I think what people have missed so far is that, if you want people to retrieve that DNA-based information in the far future, you can do infinitely better that break it down into bits according to some completely arbitrary set of encoding rules, filled with redundancies and error-correcting codes and ISO norms that make sense only if you know the technological history of the entire 20th and 21st centuries.
If you want aliens, metahumans or supersapient koalas to be able to read and enjoy your Harry Potter fanfiction, you don't leave them to struggle with a bunch of arbitrary bits. You simply give them genes, and as a decoder, bacteria to splice these genes in.
The nifty thing with DNA is: it's iconic.
You want a symbol that means light? You use a gene that *makes* light - bioluminescence.
Want to encode warm and cold? A gene for exothermic or endothermic reactions.
Up and down? A gene that causes gravitaxis.
Hopefully, combinatorics is universal enough that you can recognizably associate these genes to encode more ideas: light and heat for fire, light and up for stars.
Our most abstract ideas (equality, conservation) are generally derived from physical or social processes, which you can demonstrate with carefully coordinated sequences of genes.
In other words, writers of the eternal language would not be novelists, but playwrights. Genetic drama is our best shot at an enduring legacy. With gene drive and a lot of patience, you could even get the biosphere as a whole to store and enact your Twitter trysts and Facebook feuds, among all the weird nonadaptive mutations that creep under the radar of natural selection.
There is a theory that states that this has already happened.