To put one's thinking cap on means to take time to think something over. It likely has its origins in the 17th century when jurists and other scholars commonly wore tight-fitting, square caps. An English judge of this era would put on his "considering cap" (his white wig) before passing sentence in all cases. back then some said it was considering cap.
Follow up - A new paper in the Journal of Experimental Social Psychology finds that ”wearing a white lab coat, a piece of clothing associated with care and attentiveness, improved performance on tests requiring close and sustained attention.” The researchers found no effects when the coat was identified as a painter’s coat. “The main conclusion that we can draw from the studies is that the influence of wearing a piece of clothing depends on both its symbolic meaning and the physical experience of wearing the clothes. There seems to be something special about the physical experience of wearing a piece of clothing.”