It depends on the context and circumstances. I can think of a few things that qualify:
A black person has the privilege of being able to go to an Ivy League school with lower-than-average grades due to affirmative action.
A black person has the privilege of being taken more seriously as a rapper or other urban culture figure.
A black person has the privilege of being able to talk about race or challenge ideas bout racism more openly without being accused of 'whitesplaining' or something similar.
A black person has the privilege of being taken more seriously than a white person if they were a victim of a hate crime. If a white person is attacked, kidnapped and/or abused because of their skin colour, there are evidently many people who won't even call that racism.
There's also the fact a black person has the race card in their deck, ready to be pulled at a time of convenience.
I would also call it a privilege to have the progressive media pander to you constantly. Don't even get me started on the pandering that occurs on university campuses (see the US and Canada presently).