Why are "hard water" and "soft water" so called?
Solution 1:
Hard water contains minerals that actually harden. Over time, they form stains and crusts on the pipes and vessels they contact. If you look at the inside of pipes that have carried hard water for decades, there are thick stonelike coatings.
Soft water does not exhibit such behavior.
See a discussion of hard water deposits and scale here.
Solution 2:
Hard water contains >160ppm of minerals (typically calcium compounds) and actually feels harder when drunk than soft water does (<160ppm). The earliest use in OED below would seem to support that origin rather than it being "hard" to form a lather.
14. a. Applied to water holding in solution mineral, especially calcareous, salts, which decompose soap and render the water unfit for washing purposes.
1660 F. Brooke tr. V. Le Blanc World Surveyed 18 : The water was sharp and hard, but nothing brackish.
1756 C. Lucas Ess. Waters i. 83 : Hard waters are the best for builders and plasterers.
1805 W. Saunders Mineral Waters 305 : A very hard water, curdling soap, and possessing a large portion of selenite and earthy carbonats.
1849 R. T. Claridge Cold-water Cure (1869) 85 : Hard water makes the skin rough, but soft water, on the contrary, renders it smooth.
Solution 3:
The reasonable possibilities would seem to include:
Hard water comes from the ground, which is hard, soft water from rain. (All the more reasonable when we consider times when people had less of an understanding as to why some ground-sourced water is soft).
It's hard to get up a lather in hard water. (Conversely, it's easy, not soft to do so in soft water).
The residue left behind by evaporating hard water, is hard in the physical sense.
Vegetables, particularly hard legumes, do not soften as quickly in hard water. In particularly hard water, they may not soften at all unless you use baking soda (which results in the sort of soft flavour and nutrition-free vegetables that was so beloved of a previous generation, in some parts of the world). Blanching in very hard water can even harden soft fruit and vegetables.
While some drinks benefit from hard water (it benefits some styles of beers, and hurts others), generally soft is preferred for tea-making. It's hard to make a good cup of tea with it.
Other qualities seem less likely; that it can sometimes taste nasty but the best-tasting waters are hard, that the waters of spas long considered to have health benefits are particularly hard, that it messes up modern central heating systems.
An interesting commentary on hard water from 1751, Francis Home's An Essay on the Contents and Virtues of Dunse-Spaw argues:
To be ſoft is one of the chief properties in the offices of common life. The hardneſs of water, I find, depends moſtly, if not entirely, on ſea ſalt : for water, that eaſily diſſolves ſoap, is hindered from doing it by a mixture of ſea ſalt ; even after ſoap is diſſloved, the equal ſolution is curdled by an addition of that ſalt. As the ſame effect happens betwixt ſalt of tartar and ſea ſalt, the curdling of ſoap muſt, ſtill be owing to the action of theſe two.
The interesting thing is how wrong they are! Now, they're clearly on the right mark in noting that salt of tartar and sea salt both have the hardening effect, but he's blaming sodium chloride for the hardness common to water, and not suspecting that it could be another salt again.
He defines "soft water" thus:
The criterions of ſoft water are, that it brews well; boils peaſe without making them hard; diſſolves soap equally; and washes dirt entirely out of linnen.
As we would expect, having found he does not have a clear picture of the cause of hardness, he focuses on the effect not the cause.
It would seem from this, and other texts of the time that scaling wasn't much considered. That soft water from rain sources was used in some cooking tasks could explain this, as it would reduce the amount of scaling witnessed, but also complicate the question of which water was resonsible. This also in itself speaks to the most likely origin of the word; he was not the only person to note the effect on "peaſe", but it was well-known to cooks (and remains culinary knowledge to this day).
(Alas I cannot find references for the belief that hard water blocks "pores" in vegetables, nor the advice to soak beans in soft but cook them in hard for better colour or favour, though I'd like to as the foodie in me likes the idea of the dedication that uses both well-water and rain-water for the same item [just add some salt! Jeez!]).
In all, while I could just about credit the ground = hard possibility, it seems that the effect on vegetables is the most likely source:
Soft water makes vegetables softer on boiling and soaking than hard, and very hard water can even harden some.
Solution 4:
The earliest reference I could find (not including that Brooke reference, which I think is erroneous) is 1712 in the The Natural History of Northamptonshire by John Morton. He says:
The common springs that have a little, slight Tincture of Vitriol, which with us are usually called Hard and harsh Waters, are vulgarly known to be such by their not taking so speedy or so lasting a Lather with Soap; by their turning and curdling of Milk in case a smaller Quantity of Milk is infus'd upon a much larger Quantity of the Boyling Water; by their requiring a larger Quantity of Malt to make the Drink, that's brew'd with it, of equal Strength to that which is made with softer Waters; and by their not boyling Pease, and other Garden Stuff so quickly or so soft and tender, as do those of a milder Nature. A Person of a nice Palate may distinguish them by the Tast; And to one who washes with them, whose Skin is of a finer Texture, they are sensibly harsh, and leave a little Roughness on the Hands and Face.
Harsh water actually seems to be an earlier term for the same thing, being found as early as 1667 in The History of the Royal-Society of London, for the Improving of Natural Knowledge by Thomas Sprat (earlier if you count the l'eau aspre) and being the opposite of soft water. Perhaps hard water was derived from soft water as being more clearly its opposite.