Toddler with Water Allergy...
[livescience.com]
You would think that they would have tested if there is a reaction to distilled water. If there isn't, it would point to something in solution in the water. If there still is a reaction, then it will point to the second theory where a substance found on their skin produces a "toxic" material when it interacts with water, leading to hives. In this case, once the substance is identified, a neutralizing compound can be dissolved into the water and voila!
The distilled or purified water "test" sprang to mind for me, too. I'm surprised it wasn't addressed.