Last spring, the internet went wild for a male Bald Eagle who incubated a rock. The bird, named Murphy, is a permanent resident of a rehabilitation center in Missouri. Earlier in the year, the raptor had raised eyebrows by building a nest on the ground of his outdoor enclosure, but soon after that he did something even more surprising: He found a smooth beige stone to sit on. For the next three weeks, Murphy became increasingly protective of his rock—aggressively defending it like a real egg—until staff at the sanctuary swapped the stone with a rescued eaglet, which Murphy then promptly took under his wing. Murphy’s antics delighted his many online followers, but his seemingly odd behavior was an example of a quirky phenomenon in the avian world. Scientists have long documented gulls sitting on pebbles, geese coddling golf balls, and even shorebirds tending to round(ish) bones—objects that experts refer to as pseudo-eggs. But so far, researchers have yet to reach a...