Is Mississippi cognate with Michigan?

Calling on the U. Minn. Ojibwe dictionary, Ojibwe gitchigami = Gitche Gumee referred to in Longfellow's Hiawatha means "Big lake". Gichi means "very big". Mich is the initial which underlies michaa and other words meaning or including the meaning "big". Although the deconstruction of mich+?+gami does not seem to work in the contemporary language, it is plausible that michigami did exist meaning "big lake". (michi means "by hand, barely" so is unrelated).

The root ziibi means "river" and Misi-zaaga'iganiing (source of Mississauga) means "Great River-mouth" in some related language, perhaps, but in the Minn. Ojibwe dictionary is just glossed as a place name; zaaga'igan is "river". Combined with the fact that meʔši is reconstructed for proto-Algonkian meaning "big", it is fair to say that the source of the [mɪ{s,ʃ}] part of the words Michigan and Mississippi are probably the same root meaning "big", and probably from Ojibwe given the location of the headwaters of the Mississippi and Lake Michigan. The word for "water" is nibi, PA *nepyehs, so "big water" is not a plausible analysis. PA *si:pi:wi 'river' is found in Cree, Ojibwe, Fox and Menominee so it almost certainly means "big river". This article gives other examples and variant reconstructions of the roots kamy and sipiw.

The states themselves are only marginally related, by being American. The words are related.