George R. Stewart has a whole chapter on this general topic in Names on the Land. He says that Arkansas is the only state name “about which pronunciation and spelling ever rose to be a major issue”. Arkansas/Arkansaw, originally referring to the river, was part of the Louisiana Purchase and, like many other place names in the region, Americans adopted the French pronunciation (or at least an approximation of it). And the name was often, but not always, spelled accordingly: Arkansaw.

In 1819 Congress passed a law creating a territory spelled, according to the Act of Congress, Territory of Arkansaw. In the very same year a New Yorker named William Woodruff moved to the new capital of Arkansaw Territory. Woodruff felt very strongly about language and spelling. He even attacked Noah Webster for including the word “lengthy” in his dictionary, for clearly not being a real word. Woodruff was also a printer by trade and set up a newspaper in the new Arkansaw Territory. In the first issue he printed the Act of Congress that had created Arkansaw Territory. In this Act the word “Arkansaw” occurred eight times. Woodruff felt very strongly it ought to be Arkansas, which was still a common alternate spelling. In his newspaper’s printing of the Act of Congress he changed all eight mentions of Arkansaw to Arkansas.

The population of the territory was only about 10,000, and very many could not read. And most had little to no preference over the spelling of the territorial name. In any case, Woodruff’s newspaper had a huge influence on this particular issue. As Stewart puts it, “Apparently even Congress forgot about [their] original spelling, and later bills used Arkansas”. Reprintings of the Act of Congress quickly started spelling it Arkansas, even official, federal printings. In short, official documents “simply changed the spelling without comment, as if a mere clerical error were being corrected”.

As the years passed the people of Arkansaw Territory more and more spelled it Arkansas, yet continued to pronounce it Arkansaw. In other parts of the US there was the feeling that the spelling and pronunciation should match. And as the spelling Arkansas became common some people began to pronounce it “Arkánsas”, like the state Kansas, with ar- prepended—although the place names Arkansas and Kansas have different etymologies and the close resemblance is coincidence [ed: or not, see below]. As more time passed even some of the people of the territory began to say Arkánsas. Still, both spellings and pronunciations were fairly common.

In 1836 the territory was made a state, spelled Arkansas. Yet there was still a question about pronunciation. In the 1840s the two senators from Arkansas disagreed with each other about it. In Congress this resulted in one being called “the senator from Arkansaw” and the other as “the senator from Arkánsas”.

By 1850 the issue was not solved at all, neither pronunciation or spelling. Some gazetteers spelled the state Arkansas, some Arkansaw. Some offered both, and both pronunciations too. Sometimes the state was spelled Arkansas and the river was spelled Arkansaw. As the state of Kansas became developed and well known, it helped solidify the spelling Arkansas over Arkansaw, and for many people made the pronunciation Arkánsas common—with an accent on the second syllable. Still, the people of Arkansas still overwhelmingly pronounced it Arkansaw.

After the Civil War many people immigrated to Arkansas and brought their pronunciation Arkánsas with them. This pronunciation became common in Little Rock. Stewart says, “By 1880 anyone might have decided that Arkansaw was a dying and backwoodsy manner of speech”. However, the natives of Arkansas would not so easily change their ways, and before long the issue became important. The pronunciation, and even spelling “Arkansas” became to be condemned by natives of the state as “Yankee persecution”. Increasingly there were calls for rejecting Arkánsas.

In 1880 the legislature of Arkansas took up the issue and formed a committee to resolve things. The committee, being mostly made up of native Arkansans, was not unbiased and from the start worked for “Arkansaw”. They supported this in many ways, drawing from history, tradition, etc (though undoubtedly cherry-picking to some degree). They even cited the Yankee Longfellow, who had written “I prefer the sound Arkansaw as being more musical than Arkansas”. The committee’s final report unanimously recommended the pronunciation Arkansaw. I’m not sure if they said anything about the spelling, which by then was rather firmly established as Arkansas.

In 1881 the state legislature passed an official resolution declaring the pronunciation to be “Arkansaw”, in accord with the local manner of speech. They also declared the pronunciation Arkánsas “an innovation to be discouraged”.

For most of the country, this settled it. The state was spelled Arkansas but pronounced Arkansaw. So be it. Except! There is also the Arkansas River, which flows through a great deal of Kansas. The people of Kansas tended to call the river the Arkánsas River. And there was a town by the border called Arkánsas City. After the state of Arkansas declared the pronunciation Arkansaw, and the rest of the country said “okay”, dictionaries soon began to say the river was also pronounced Arkansaw, as was Arkansas City, in Kansas. Some people of Arkansas tried, mostly half-heartedly, to convince the people of Kansas to pronounce the river and Arkansas City “arkansaw”. But since they had called upon local usage tradition in their own support of “arkansaw” they could not easily say that the people of Kansas had to abandon their local pronunciation traditions.

Arkansas City, Kansas, is still pronounced Arkánsas. And in much of Kansas the Arkansas River is likewise pronounced Arkánsas. The same goes for Arkansas Street, in Wichita, Kansas.

But to answer the question directly and tersely, the spelling of Arkansaw Territory was changed to Arkansas mainly due to the efforts of William Woodruff, who started the first newspaper in Arkansaw Territory but always published the name “Arkansas”. Congress didn't seem to care, or even realize, possibly, that they had named it Arkansaw but soon began spelling it Arkansas. As for why the spelling and pronunciation don’t match, it is because both come via the earlier French place names—the same reason the same is true for Illinois.

Names on the Land, George R. Stewart.

edit PS: Stewart might be wrong about the etymology of Arkansas and Kansas being unrelated. William Bright, in Native American Placenames of the United States says they do derive from the same Siouan root, but there is some additional complexity I don't have time to get into at the moment. One bit though is that Arkansas comes from a French interpretation of an Algonquian language term, which involves a Siouan term within it.

(edit: tpyos, words)