Exhausted from anger, strain, or other great effort. For example, You can argue until you're blue in the face, but I refuse to go. This expression alludes to the bluish skin color resulting from lack of oxygen, which presumably might result from talking until one was breathless. See also under talk one's arm off.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.