The river neither starts there (Minnesota) nor ends there (Louisiana), nor does it seem to even have the longest border with the river (looks like Illinois does?), so why is that particular state named after the river? I'm assuming just because of some kind of timing issue, but I could not find a real answer anywhere.
Your assumption that it is an issue of timing is correct.
I started an answer trying to explain the history of colonial borders and westward land claims, but got lost in the weeds. So, I'll try and get directly to it.
this map gives a general overview of what territory the United States and the individual states claimed after the Treaty of Paris that ended the Revolutionary war. The key area to pay attention to is that magenta colored rectangle of territory bordering Georgia and Spanish West Florida marked "disputed territory".
Eventually, after several years of negotiations between the United States and Spain, Pinckney's Treaty was signed in 1795 and that disputed territory was ceded to the United States.
That territory was organized by the federal government as the Mississippi Territory in 1798. The state of Georgia continued to lay claim to the lands that now make up the northern half of the modern states of Mississippi and Alabama. However, in 1802 the details of the Yazoo land scandal started to come out, and the state of Georgia ceded its western claims to the federal government, which was then incorporated into the existing Mississippi Territory. The coastal strip of West Florida would be ceded to the United States in 1812.
Because of the rich agricultural land in the Mississippi delta, that region of the territory attracted Anglo settlers a little bit quicker than territory than regions to the east, so in 1817 the territory was split in half, with the west gaining statehood as the state of Mississippi. The less populated eastern region became the territory of Alabama, and gained statehood two years later in 1819.
So, why was this territory named Mississippi and not the other areas you mention? Well, in 1798 the areas of Minnesota and Illinois were still part of the Northwest Territory, and was still heavily populated by Native American peoples. Indeed, the Battle of Fallen Timbers in 1794 broke Native American resistance to Anglo settlers pushing into Ohio. So, it is only in 1800 that enough white settlers move int the region to force an administrative reorganization to split Northwest Territory into the Ohio Territory and Indiana territory. Prior to 1798, there were not yet enough white settlers in the region to force the federal government to reorganize into smaller territories and necessitate a new geographical name.
Louisiana did not come into the United States until the Louisiana Purchase in 1803. In 1798 it was still Spanish territory.
For more reading about the early Mississippi Terrritory, I recommend checking out The Mississippi Territory and the Southwest Frontier: 1795-1817 by Robert Haynes.
