How much plain water should my 6 month old drink a day?

Should=none. Plain water is not considered a needed part of the infant diet. Breast milk or formula are all they need. On occasion, it is a reasonable alternative to other fluids, and would be much better than juices.
Plain Water. Usually at 6 months of age, we start to introduce the sippy cup. We usually suggest introducing the sippy cup with the use of water. Children at this age do not really require water so we only use it as a supplement. Infants generally get adequate hydration from their breast milk and/or formula, if they are getting their required amounts somewhere between 20-28oz per day.