There is no "pure canon" equivalents of pulling the sword from a hat, however...
Slytherin - be able to control a Basilisk.
The qualities which Slytherin prized in his "handpicked students" included resourcefulness, determination, and a certain disregard for the rules, along with the ability to speak Parseltongue. He also selected his students according to cunning, ambition, and Blood purity
Ravenclaw - pass an IQ test with a minimum score of X. Or at least answer a difficult riddle.
Ravenclaw is described by the Sorting Hat as having selected students according to intelligence and wisdom.
Hufflepuff - either fail to be accepted in any other houses :); For specific high level task, achieve something high level with a lot of hard work without having significant abilities for the task (e.g. manage to grow Mandrakes to un-stone people).
Hufflepuff admitted students into her House that were unafraid of hard work. While the other Founders took students based on either ambition, bravery, or intelligence, Helga Hufflepuff took the rest, treating them all equally and teaching them all she knew