What approach should be taken to writing a one-minute speech about an unusual career option for an oral exam?
As a one minute speech is extremely short, constituting no more than a single page of text, an ideal is a very narrow focus. You should, for your chosen career, discuss why the career appeals to you, the salary, the skills needed, the sorts of things you do on the job, and employment opportunities. As you choose your career for the speech think about what might make the career unusual -- subject matter, type of activity, place, employer, etc. For example, light house keepers and polar explorers work in unusual locations, herptologists work with amphibians and pet morticians work with deceased animals.