Deionized water is the best bet (better than softened water), but making it in large quantities can become a little pricey. D.I. water is what auto factories use in their paint operations, both as the carrier liquid for water based phosphate and electrocoat prime systems and for the final rinses prior to baking and subsequent spray prime and top coat operations. You may be more familiar with it as the "Spot Free Rinse" at your friendly car wash.
Resin bed filters are available that will make small volumes of deionized water, and in fact that is what the Mr. Clean system uses. Unfortunately, the tiny flow rate and short life of the Mr. Clean filter probably isn't the answer for those with serious mineral problems. Try a web search for deionized water, you'll find more than a few companies that sell larger systems that would be more practical than the Mr. Clean product. You'll want to start with a good sediment filter first, then the D.I. water filter so the sediment doesn't clog the resin bed. Considering the cost, you might also want to set it up to only use the D.I. water for the final rinsing operation.
Jerry