"She had said goodbye too many times before" means, at some point in the past, it was the case that she had previously said goodbye too many times. I think this is the intended meaning.
"She said goodbye too many times before" I don't think makes sense if you're extremely literal about it, since I can't see what "before" would track to without the embedded past.
The grammatically correct versions I can come up with are: - She had (or she'd) said goodbye too many times before. - She said goodbye too many times. - She said, "Goodbye too many times before".
Disclaimer: I do get that these are all worse song lyrics and that nobody had any problem understanding the intended meaning of the example sentence, which is sort of the goal of grammar.