Apollo is the Greek and Roman god of the sun, and is just inside the top 1000 names for boys. It's also the name of the legendary theater in Harlem—and Rocky's biggest nemesis in the classic boxing movies.
Their son's two middle names are the maiden names of Stefani and Rossdale's mothers. But Bowie is often chosen in homage to legendary singer David Bowie.
Flynn has become a hot name, currently in the top 700 and rising. He was the hero in Disney's Rapunzel flick Tangled, and it was the name picked by Orlando Bloom for his son.
The name is a bit of a departure for the Stefani-Rossdale clan, as the other two boys' names are place names, but the name fits in pretty well with the rest of the clan.