Oh, how I love trivia like this. Jesse Sheidlower explains in Slate the linguistic source of Sarah Palin's accent.

I mentioned the accent a while back in a post about Tina Fey's first SNL skit: "Although it's not at all clear how Palin comes by her northern-plains accent, Tina Fey does a superb job in the skit of emphasizing it just enough to highlight the Fargo-like absurdity in it all."

Sheidlower makes it very clear how she comes by the accent:

Overall, because of the mixture of people and the large number of newcomers, Alaskan English is often hard to place, with both Westerners and Midwesterners thinking that it sounds oddly foreign; indeed, some Westerners have said that Palin sounds like a Midwesterner, and Midwesterners that she sounds Western.

Others have wondered whether her accent hails from Idaho, where her parents are from. But dialect features tend to come from one's peers, not one's parents, and Palin spent her childhood in Alaska's Mat-Su Valley, which is where she got her distinctive manner of speaking. The next town over from Wasilla, Palmer, has a large settlement of Minnesotans—who were moved there by a government relief program in the 1930s—and features of the Minnesotan dialect are thus prominent in the Mat-Su Valley area. Hence the Fargo-like elements in Palin's speech, in particular the sound of her "O" vowel. (Despite its name, Fargo took place mostly in Brainerd, Minn.) However, even in the area, many people speak a more general Alaskan English, the sort one would find in nearby Anchorage. Palin's frequent dropping of the final G in -ing words and her pronunciation of terrorist with two syllables instead of three are characteristic of general Alaskan English (and Western English) rather than the specific Mat-Su Valley speech.

Since I grew up in Montana and have lived most of my adult life in Washington, with a couple of stays in Colorado along the way, I'm definitely a Westerner. (And, yes, I too drop "g"s and will probably sometimes substitute "ya" for "you".) And I'm also among the "some Westerners [who] have said that Palin sounds like a Midwesterner."

Now, if only someone would explain how W. came up with that Texas-like accent when his brother Jeb who spent more time in Texas didn't pick it up.

Source: Placing Sarah Palin's accent. - By Jesse Sheidlower - Slate Magazine