In early colonial times a lot of European explorers recorded Native American Indian (I'll just say "Indian" after this) names for places and these became a normal part of the named American landscape. In later colonial times and through the Revolutionary era Indian names generally fell out of fashion, as a lot of settlers thought the Indians savage, ignorant, treacherous, heathen, etc.

But by the early 19th century large parts of the settled eastern part of the US no longer had much direct experience with Indians and older ideas of Indians and Indian names being "uncouth" dwindled. At the same time there was a growing popularity and romanticized notion of the "noble savage". Indian names began to become popular, especially for counties, at first. Into the early 19th century Indian names became increasingly popular and "fashionable". New states and cities increasingly took Indian names. Sometimes not for fashionable reasons but just because the name had been in common use for so long it seemed perfectly normal (for example "Chicago").

On the frontier, where direct contact with unsubjugated Indians still happened, Indian names remained less popular. Meanwhile in, for example, Massachusetts, the names became fashionable more quickly. The first Massachusetts town to take an Indian name on purpose was Seekonk, in 1812, then Saugus in 1815. But it wasn't really until the 1840s that the fad really caught on in a big way.

Most Americans at the time, and away from the frontier, did not care much about tribal distinctions or where a place name was "supposed" to be. People found nothing strange in transferring an Iroquois name, say, to somewhere far away like Louisiana. Sometimes it didn't even matter if the names had actual Indian origins or not, so long as they sounded Indian. For example a lot of the counties of Michigan have Indian-sounding names, but a large portion of them were simply invented to sound Indian, by Henry Schoolcraft. Another example: When Pasadena, California, was established the town founders wanted a fashionable Indian name but did not care in the least whether the name had anything to do with southern California. They had a friend who was a missionary among the Ojibwe in Minnesota, and through him they eventually settled on Pasadena.

The fad for Indian names was strengthened by people like Washington Irving, Henry Wadsworth Longfellow, and Walt Whitman. This fad for Indian place names remained strong through most of the 19th century, although usually only after direct contact with Indians had become just a memory in a particular region.

That is the basic answer, mostly taken from George R. Stewart's Names on the Land. To address the specific names you mentioned:

The use of "Miami" in Florida has different origins from the use in Ohio and the Midwest. In the Midwest the name usually comes from the Ohio tribe whose name was Anglicized as Miami (sometimes "Maumee" and other variations). In Florida the name came from the Mayaimi tribe of Florida, through Spanish to English. We spell them the same now, but they have unrelated origins.

"Wyoming" was originally the Anglicized name of a valley in Pennsylvania. In 1809 Thomas Campbell published a romantic and sentimental poem called Gertrude of Wyoming. The poem became very popular and as a result the name "Wyoming" was applied all over the place. No one really cared if it made sense to use the word in one place or another. It was simply a famous and popular Indian place name during a time when Indian place names were "fashionable".

I don't know the details about the history of the use of "Wabash".

(edit: tpyos and words)