"The musical name "Wyoming" was used by J.M. Ashley of Ohio, who, as early as 1865, introduced a bill to Congress to provide a "temporary government for the territory of Wyoming." It was to be formed from portions of the Dakota, Utah and Idaho territories. The bill was referred to a committee where it lay idle until 1868. During debate on the bill in the U.S. Senate in 1868, other possible names were suggested, such as Cheyenne, Shoshoni, Arapaho, Sioux, Platte, Big Horn, Yellowstone, Sweetwater and Lincoln. "Wyoming" was already commonly used and remained the popular choice."
The word was already known (to white people) in the US at the time as Wyoming Valley, Pennsylvania, which was the scene of an "Alamo" type massacre during the revolution. It was rather well known at the time in the US.
I've seen a zillion translations of indian names for stuff that are all over the place. Depending on where you Google, the word Wyoming either means big prairie, windy place, place between mountains, river bottom, or a van down by the river.
Ok I made that last part up. But I bet some tribe out there might have said it.