According to space.com (cited by Wikipedia), Edwin Eugene Aldrin Jr. "took [Buzz] as his legal name" in 1988. Question setters need to be aware of this, as it means that although his 'birth forenames' are Edwin and Eugene, his 'real' forename is actually Buzz.
The same sources tell us that he came to be called Buzz because in his infancy, the closest his elder sister Fay (18 months his senior) could come to saying "brother" was "buzzer". The nickname obviously came in handy, what with the names Edwin and Eugene already being used to refer to the children's father.
© Haydn Thompson 2020