An archaic Hogwarts term for any new student whose Sorting takes longer than five minutes. This is an exceptionally long time for the Sorting Hat to deliberate, and occurs rarely, perhaps once every fifty years.
Of Harry Potter’s contemporaries, Hermione Granger and Neville Longbottom came closest to being Hatstalls. The Sorting Hat spent nearly four minutes trying to decide whether it should place Hermione in Ravenclaw or Gryffindor. In Neville’s case, the Hat was determined to place him in Gryffindor: Neville, intimidated by that house’s reputation for bravery, requested a placing in Hufflepuff. Their silent wrangling resulted in triumph for the Hat.
The only true Hatstalls known personally to Harry Potter were Minerva McGonagall and Peter Pettigrew. The former caused the hat to agonise for five and a half minutes as to whether Minerva ought to go to Ravenclaw or Gryffindor; the latter was placed in Gryffindor after a long deliberation between that house and Slytherin. The Sorting Hat, which is infamously stubborn, still refuses to accept that its decision in the case of the latter may have been erroneous, citing the manner in which Pettigrew died as (dubious) evidence.