My mother once told me that there was a time (before at least the 1980's) when people did not use the word "Cancer" when discussing the disease. What was the reason for this taboo in the US?

As far as I can tell the premise of your question is flawed. I haven't been able to find anything that shows there was any national taboo surrounding the word "cancer" prior to the 1980's. Rather I've found several sources from the era in question that do use the word. Edit: Example #1, Example #2

This is not to say that there has never been taboo surrounding cancer, or that euphemisms have never been employed to reference it. Simply that I can find no evidence that there was a widespread national taboo on the word prior to the 1980's.