Do they have to choose a Saint's name?
No. They can choose the name of a previous pontiff. However, this is very much like the Sacrament of Confirmation, when catholic youth are 'confirmed' in the faith. The recommended formula is to choose a saint's name and model their life after that saint's qualities.
This tradition began back in the early centuries of the Church when the newly elected pope had been named after a pagan god - Mercurious. He chose a new name that was more faithful to his beliefs. These days, a new pope often picks the name of his favorite saint or pope to be his papal name.