Balancing on one leg

Is there any significant benefit to doing this while doing daily activities? (While brushing teeth or doing the dishes or any other activity where you stand in one spot)

If you can't do a quad stretch without hopping all over the place while balancing on one leg then you should do this.

works the nervous system and small stabilizing muscles.
for added difficulty stand on a pillow and make sure the arch of your foot doesn't hold any weight on it. (helps with pronation problems and is good for ankle related injury preventation).

I suppose you are less likely to get an ankle injury if you do this a lot?

I think I mentioned that in my above post ...
but just balancing on one leg is too easy and does not load your nervous system enough to be the beneficatory..
balancing on one leg doing squats with it on the other hand, or while drawing circles with the other leg, or joggling 2-3 balls in the air with your hands.

i do one legged squats on a bosu ball.