How Styx Got Their Name
For a time in the 1970s, there was no escaping progressive rock, or "prog rock," as it was sometimes called. Musicians who recorded and performed prog rock — such as Kansas, Yes, and Emerson, Lake and Palmer, among others — wanted to elevate rock 'n' roll from its basic structures into something more elaborate, combining elements of classical and other genres of music, as Prog Archives explains. As a genre, it's mostly niche nowadays, but some of the biggest songs in the history of rock, such as "Carry On, Wayward Son," by Kansas, came from prog rock.
Styx was, at its core, a prog rock band, although at the same time, the band sold out arenas with its anthems and power ballads. This dichotomy between being a group of face-melting rockers and storytelling musical artists helped lead to the band's downfall, but for the better part of a decade, there was no bigger name in rock than Styx (per Louder Sound).
The Tradewinds and TW4
Chicago in the late 1960s and early 1970s was the breeding ground for several acts that would later make it big in rock, including Chicago Transit Authority (later just Chicago). Two suburban brothers, John and Chuck Panozzo, hoped to make it big in rock and started a band they called The Tradewinds, as Louder Sound reports. With John on drums and Chuck on bass, the two men formed the rhythm section, but they still needed a vocalist and guitarist.
The brothers eventually brought on Dennis DeYoung, and later learned that another band going by the name The Trade Winds was making it big, so they needed another. They settled on TW4. By this time, the band was getting big performing in clubs, and they began looking for a record label. After solidifying the lineup even further, including by signing James "JY" Young and John Curulewski, the band signed on with Wooden Nickel Records, but their new managers asked for a name change.
The river of the dead
If you're going to have a prog rock band, it doesn't hurt to lean into the mythological, the mystical, and the arcane, both as sources for your lyrics and for the art on your album covers. So, it's a good job the word "Styx" comes from Greek mythology.
As Britannica notes, Styx was the name of one of the underworld's rivers in Greek mythology, a barrier between the living and the dead. Whenever gods swore an oath, they would do so by the river's waters. And, when the hero Achilles was an infant, his mother grasped him by the heel and dipped him in the river's waters, making him immortal, save for in the spot where she held him. Incidentally, it was an arrow to the heel that killed Achilles.
Despite the deep mythical and poetic import contained in the river and the legends surrounding it, it seems that John and Chuck Panozzo, Dennis DeYoung, and the rest of the band only landed on that name because absolutely nothing else was working for them. DeYoung said in an interview with Circus Magazine in 1979 that the name Styx came up in the discussion of names, and they kept it because "it was one of the hundred names we tried, and it turned out to be the only one that none of us hated," (via Great Oldies).