Elias Story is the identity Isolt Sayre assumes when fleeing from her aunt, Gormlaith Gaunt, and sailing to North America on the Mayflower. Elias is a Muggle boy who fits in better with the shipful of Puritans than a witch would (Pm).
Around this time, Isolt cuts her hair short as part of her efforts to evade her aunt, but it is unclear whether she does so before or after deciding to masquerade as Elias Story (Pm).
A boy or young man named Elias Story really is recorded as a passenger on the Mayflower (source: Wikipedia).
Rowling probably selected that name from the list of passengers for the reasons below.
Elias = similar to "alias"
Story = as in made up, fictional