In English we often say, for example, "he still has a ways to go before he's done." Is this grammatically correct?
AHD4 has the following entry for ways:
ways n. Informal (used with a sing. verb) Variant of way. See Usage Note at way.
The Usage Note states, in part:
In American English ways is often used as an equivalent of way in phrases such as a long ways to go. The usage is acceptable but is usually considered informal.
As an idiom, the phrase seems well-established in the United States. As Peter Shor said, Ngram shows evidence of this. The Corpus of Contemporary American English gives me 138 hits for "a ways to go" and 193 for "a way to go". Many of these hits are from published print sources. (In the British National Corpus I get no hits for "a ways to go".)
As a native English speaker, I think "a ways to go" sounds fine.
EDIT: Also, see page 949 of the Merriam-Webster's Dictionary of English Usage.
"Ways" is not the plural of "way" but a separate word with its own meaning. "Ways" means "a certain distance, not great, perhaps, but significant"; it is often modified as "a little ways," which makes it mean "a very moderate distance." The distance may be physical or figurative. "Ways" is singular, always seen as "a ways." On this understanding, your usage is grammatically correct.
"A ways" is just how some of us Americans talk and with never a second's doubt. Hear Harvard man EJ Dionne on NPR. Doesn't know any better.
Other Americans talk that way as a joke, to affirm hick roots or to be deliberately rustic as the English commenter notes.
Would one expect more, when some of use plural noun, "savings", as a singular subject? Not when many ads say something like this. "Buy one, get one free! Now that's a savings."
Many do the same with "an Olympics" meaning "games". People who talk that way are not nimble enough to say "these Olympic games" or this "Olympic game series" taking a singular verb because "series" is the subject.
What prompts these solecisms? Local composers of cheap advertisements. NPR announcers, who barring a few, are not all that educated.
For all their loud, hectoring, over-the-top ranters like Limbaugh and Bill Reilly apparently got some education along the way.