"I’d say that the rumours of Hogwarts’s beloved Scorpion King losing his mind are well founded."
-- Severus Snape (CC3.5)
After Albus Potter and Scorpius use the Experimental Time Turner, Scorpius finds himself alone at Hogwarts, where he has a different personality and also a reputation as the leader of those students who consider that Muggle-borns are fair game for torture and abuse (CC3.2, CC3.4, CC3.5).
Karl Jenkins first calls Scorpius by this title, when he and another member of their Mudblood-baiting gang meet in the grounds of Hogwarts. Another of these students, Polly Chapman, also uses it when she tells Scorpius she would like to be asked by him to go to the Blood Ball (CC3.2).
Craig Bowker Jr also says that The Scorpion King "hates homework", offering to do it for him when they are in the School Library (CC3.4) and Severus Snape says it when referring to rumours going around the school (CC3.5).
It finally becomes (almost) an affectionate name, when Rose Granger-Weasley uses it after all the loose timeline strands have been put back into place (CC4.14).
The title for Scorpius as the student leader (king) is a play on his name "Scorpius", which is derived from the constellation and an extremely dangerous stinging insect (Oxford English Dictionary).
As Rose wasn't involved correcting the events from the alternate timelines, she has probably heard the "Scorpion King" nickname from her mother or father (CC4.14).
From the Web
About the play: http://www.harrypottertheplay.com