The French have a lovely phrase "l'esprit d'escalier" - which literally translates as "staircase wit". It's a phrase that refers to that belated moment of inspiration that strikes as you wander down the stairs leaving a party - be it coming up with a pithy retort to an insult or a witty reply to a question. The Germans have the same phrase - "Treppenwitz". I think we need a similar phrase in English for the moment of brilliance that takes you as you scrub your back - I suggest "bathroom clarity", but feel free to make alternative suggestions.
Actually, sitting down with a cup of tea has much the same effect on me (and I suspect, most English people) - perhaps "teapot clarity" would be just as appropriate.