Use of "a ways" goes back many decades in U.S. usage. John Bartlett, Dictionary of Americanisms (1848) has this entry:
WAYS, for way, distance, space. A very common vulgarism. {It's only a little ways down to the village. —Margaret, p. 123.}
Likewise Frank Vizetelly, A Desk-Book of Errors in English (1906) says this:
ways, for way: In the sense of "space or distance," the erroneous form ways, for way, is often used colloquially, perhaps originally through confusion with the suffix -ways; as, "The church is a long ways from here," which should ne "The church is a long way," etc.
Modern commenters on grammar and usage avoid terms like "vulgar" and "erroneous," but you can still detect something less than hearty approval for "a ways" in comments like this one from Bryan Garner, Garner's Modern American Usage (2003):
way(s). In the sense "the length of a course or distance," way is the standard term {a long way}. Ways is dialectal. So it's surprising to find ways in serious journalism—e.g., "This is premature of course; Fox still has a ways to go [read some way to go?] before it's a full-fledged network." ... Newsweek, 6 June 1964, at 46.
Kenneth Wilson, The Columbia Guide to Standard American English (1993) has a fairly balanced (and, I think, accurate) take on the usage:
ways (n.) is the plural of way, and in many noun uses it is perfectly unexceptionable: There were three ways to get home. In some ways she seemed very poised and mature. Used instead of the singular, however, in phrases such as a little ways down the street, and as a suffix added to any, as in Anyways, I was too late, ways is Nonstandard, and many Standard users consider it a shibboleth inappropriate for any Standard use.