The sport is named after a school, and the school after a town. The famous story about how the game began was first told more than fifty years after the fact, by someone who did not see it.
Etymonline records "rugby" for the game from 1864, after Rugby School, where it was played. The school took its name from the town of Rugby in Warwickshire. The town appears as Rocheberie in the Domesday survey of 1086. That probably meant the fortified place of a man called Hroca, though the first part may instead be Old English hroc, "rook."
The ending has its own story. Etymonline says the Old English burh, "fort," was replaced in the 13th century by Old Norse -by, "village," probably through Danish settlers. Without that swap, it suggests, the town might now be Rockbury.
Then there is William Webb Ellis. The legend says that in 1823 this Rugby pupil caught the ball during a football match and ran with it, breaking the rules and creating a new game. Wikipedia notes that no first-hand evidence supports it.
The story's only source is Matthew Bloxam, a former pupil. In a letter to the school magazine The Meteor on 10 October 1876, he said he had heard the change began with a boy named Webb Ellis. That letter placed it in 1824, after Ellis had left. A second letter in 1880 moved it to 1823. By then Ellis had been dead for years, having died in Menton, France, in 1872.
In 1895 the Old Rugbeian Society looked into it. Its sub-committee could not find any first-hand evidence. Most rugby historians now treat the tale as an origin myth. Writing in 2005, Eric Dunning and Kenneth Sheard argue the school embraced it to claim authority as the sport split into rugby union and rugby league. The same inquiry heard Thomas Hughes call a later pupil, Jem Mackie, "the first great runner-in" around 1838.
The myth still has a trophy. The winners of the Rugby World Cup lift the Webb Ellis Cup.