The reason that distilled water becomes acidic is because purified waters are more solvent. They try to react with things to reach saturation. In this case it is with the air containing carbon dioxide. As the water absorbs carbon dioxide from the air it forms carbonic acid. This is one of the main problems labs have with using ultrapure (type 1) waters. These waters are so pure that they rapidly go acidic from the absorption of CO2. They also tend to grow bacteria very quick, which is why type 1 water is constantly being recirculated through the filters.
This is also the reason that I don't drink purified waters without first adding something to it to reduce the solvency. Otherwise the water will not only remove toxins, but also vitamins and minerals as it absorbs these substances in an attempt to reach saturation.
I personally use purified silica to saturate the water, but it can be hard to obtain. Trace mineral drops or a little trace element salt can also be added to the water to reduce its solvency. And the minerals can help maintain the pH you are looking for.