You need to be careful drinking DeIonized or Distilled water. Because it lacks ANY dissolved chemicals it will leach chemicals OUT of your body. If you cook with it you will be fine because you are adding plenty of goodies in. Even Tea or Coffee is good. But if you count on this as your drinking water then you should add minerals. It is easy to do, just add sea salt, about a quarter teaspoon per quart. You will hardly notice any taste.
However there is a new danger to think about with sea salt. Any sea salt extracted from the Pacific Ocean is out due to Fukashima Japan's nuclear release problem. Add to this that any sea salt extracted from ANY ocean today is going to include the pollution that we, humankind, have been adding to the oceans for the last hundred years or so. This is especially true for the mediterranean sea since it is kind of land locked and we, humankind, have been polluting it for almost 2000 years. Granted that was mostly just sewage until 100 years ago when we got more serious about it.
So that leaves us with the ancient deposits such the Great Salt Lake, Utah, and the Himalayan deposits (India, Tibet, Pakistan... ). There are some other underground, protected deposits such as under Detroit and Chicago but these are usually purified before reaching market to just sodium chloride with a little added Iodine.
The blood that flows through our veins is almost chemically identical to sea water with the main difference being that our blood is composed of various organized structures, red blood cells, white blood cells, etc.
So yes, sea salt contains all of the essential minerals and in just the right ratios.