People called the hat a Stetson for a plain reason: it said so inside. Wikipedia notes that the company name was embossed in gold on every hatband. Buyers read the label and used it as the noun.
John B. Stetson was born on May 5, 1830, in Orange, New Jersey, one of twelve children. His father, Stephen, was a hatter and taught him the trade. In the late 1850s he was diagnosed with tuberculosis, and a doctor predicted he did not have long to live. He left hat-making and went west.
The trip shaped the product. He moved through Illinois and St. Joseph, Missouri, then joined a party bound for Pike's Peak in Colorado and panned for gold. Along the way he turned animal pelts into felt for waterproof tents and hats. Wikipedia says a bullwhacker paid him five dollars for one of those hats. That sale gave him the idea for a business.
In 1865 he settled in Philadelphia and founded the John B. Stetson Company, making hats for Westerners. Etymonline gives the same city and year. His signature design was the "Boss of the Plains." It had a flat brim, a straight-sided crown with rounded corners, and four-inch crown and brim. It was light, water-resistant and natural in colour, and it was first sold in Central City, Colorado.
The surname soon outran the company. Wikipedia, citing the writer Bender, says that within a decade the name had become "synonymous with the word 'hat'" across much of the West. Etymonline records "Stetson" in English as a trademark name for the hat from 1902.
A related cowboy phrase has a myth attached. The "ten-gallon hat" never held ten gallons. Wikipedia, citing a dictionary of the American West, traces "gallon" to the Spanish galón, "braid," with ten referring to braids on the hatband.
Stetson died on February 18, 1906, in DeLand, Florida. A local academy he supported had already been renamed John B. Stetson University in 1889. The family name now sits on a university and on a whole category of hat.