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. 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.
