Why does American English pluralize certain singular nouns?

I generally consider "saving" to be the act and "savings" the resulting product of that act. This pundit/economist makes a similar point.

So I might say "I made a savings of $10." or "I am saving $10 by switching vendors."

I don't consider savings plural, but a collective noun. This is similar to

"One billion dollars is a lot of money."

Savings certainly seems to originate from saving. Both British and American English have a tendency to let uncountable nouns take plurals if there is a manner and contextual reason to discretize them. ("look at all the water!" for a natural body vs. "look at all the waters!" for, perhaps, an impressive number of glasses of water) As others have pointed out, it is reasonable to conceptually divide your overall saving into many saving-s, especially if they are arriving in the form of many meager paychecks. This explanation isn't entirely satisfying, though, as the word is actually interpreted as singular in current use.

Etymonline attests savings to 1737, and the compound savings bank to 1817. Usage in compounds may affect the distribution across AE/BE. American English generally requires the attributive element in a compound noun to be singular, whereas BE has occurrences of both plural and singular. From Wikipedia:

The plural may be used to emphasise the plurality of the attribute, especially in British English but very rarely in American English: a careers advisor, a languages expert.

As such, Americans may have been reinterpreting savings in savings bank/account/deposit/etc as an independent singular word, while Brits saw it as a plural form linked to the verb save. (Here I have two identical words in my lexicon: one for the derived plural savings = repeated actions of saving, and one for the independent unit savings = a sum of reserved money) The matter is further muddled by AE/BE differences on collectives. American English is unable to use bare collective nouns as plurals, and so constructions of the savings is... and similar tended to reinforce the singularity of the term. Another factor is blocking by alternate terms - namely, Wikipedia claims that the term savings and loan is uncommon in the UK, as building societies were a more common, related institution.

This whole conjecture (in its wild and unverified glory) reminds me of Steven Pinker's word structure theory, as described in "Words and Rules: the Ingredients of Language". He offers situations like bigfoot and walkman, where the term is notably distinct from its head word (foot and man), and so must be considered a single, new unit when constructing plurals. Hence most people will say bigfoots and walkmans rather than bigfeet and walkmen.

Ways in expressions like a ways off was likely not a plural form; from OED:

In a good, great, little, long ways, and a ways, the origin of the use of ways for way is obscure. It might possibly have arisen from the analogy of phrases containing the adverbial genitive.

As for why the expressions are more common in AE, your guess is as good as mine. The rise and fall of usage is subject to so many intervening historical factors, it's amazing to me that anyone can get any research done at all in etymology.

NOAD gives this usage note:

Use savings in the modifying position (savings bank, savings bond) and when referring to money saved in a bank: your savings are fully insured. When speaking of an act of saving, as when one obtains a discount on a purchase, the preferred form is saving: with this coupon you will receive a saving of $3 (not a savings of $3).

So much for the prescriptive point of view. When you hear "a savings" it is normally from ad-speak, which is a notoriously unreliable source of grammaticality in English. Nevertheless it is pervasive (which in pop culture works with the same effect as persuasive). People hear a thing often and adopt it in their own speech. Today, I doubt many people are even sure which is appropriate, and just go with whatever sounds right to their ear.

It is just one of those dialect things. Interesting exactly the reverse is also true. In the UK kids learn maths in school, whereas in the US kids learn math. There is even less excuse for that than "a ways off", since math(s) is only half of the word mathematics.