Where did you get those lyrics from? I'm not saying they're wrong; it's just best to be aware that some pretty odd transcriptions find their way onto the Internet.
Assuming they're right, I'd guess that 'the New York Times' effect on man' is the effect of being kept aware of everyday concerns. The film Saturday Night Fever, which featured that song, was all about a workaday existence relieved by a Saturday night on the dance floor
As to the meaming of 'BeeGees', I'm sure they have fan sites on the web somewhere that would explain this sort of thing in detail. My first guess is that it's a straightforward reference to the name of one of the brothers who make up the group - Barry Gibb; or possibly Brothers Gibb.