Meaning of 白 in words for "nonsense"?
One of the meanings of 白 is, roughly "in vain":


  没有成就的，没有效果的：～忙。～说


I think it's easier to imagine how white/plain become "without effect" than it is to imagine how it became "nonsense".

P.S. 白话 means "plain speech" = vernacular, as far as i know.