By googling various versions of the pun, I eventually found the story We'll return, after this message by John Walker, better known as a programmer and a co-founder of autoCAD. He maintains a website named fourmilab, where among other things, he posts short science fiction stories. I must have visited the site before, because several of the other stories were very familiar.
I had most of the main details correct. The story deals with a programmer named Art Crane, and his scheme to find a message from extraterrestrials by combing through online scientific data. The key phrase (for me) was
“The facts, dear Clifford, are not in our stars, but on our shelves.”
he said, “Why don't we look there?”
The company he and the narrator work for has a tradition of an "Annual Week of Rest" at Christmas, when "programmers would use a week devoid of managers, marketeers, meetings, and memoranda to try to out-do one another in huge bursts of concentrated effort to impress each other". They set a program scanning data on the internet, looking for patterns that would constitute an extraterrestrial message, and just before the search ends they encounter an image:
The figures on the right were humanoid, but not human. They were
simply the best aliens I'd ever seen: there wasn't a single missing or
disproportionate characteristic about them, but they were all wrong to
be humans. There were two adults, male and female—the female was
holding a little one. On the left was what was clearly a conversion
table between binary numbers represented by “−” and “|” and eight
digit symbols. The middle of the picture was divided into four boxes,
three with a large central dot, curved lines, and nomenclature using
the eight digits. The fourth contained what could only be an orrery
presentation of a planetary system. At the bottom were four rows of 64
of the digits given at the left of the picture.
They found the image by accident, being sloppy in the choice of which databases to search.
If I'd been more selective in choosing data to scan, we'd never have
found it. There could be no doubt; the Message was real, and it was
right smack in the middle of where it most definitely didn't belong.
We had found it in the NIH Human Genome database.