if sth such as a name is picked out of a/the hat , it is picked at random from a container into which all the names are put, so that each name has an equal chance of being picked, in a competition, etc.
OUT OF A / THE HAT
Meaning of OUT OF A / THE HAT in English
Oxford Advanced Learner's English Dictionary. Оксфордский английский словарь для изучающик язык на продвинутом уровне. 2005