From what I've seen the name Michigan for the territory/state came first, and was not the name of the Lower Peninusla. Michigania, being a Thomas Jefferson proposed name for a state located in what is now northern Illinois and souther Wisconsin (with an eastern border on Lake Michigan). James Monroe though is who decided to scrap Jefferson's plan and decided- north of the Ohio River either three states (super-sized Illinois, super-sized Indiana, and super-sized Ohio all extending due north of their current locations) or 5 states roughly our current ones with minor changes. In the 3 state version the Lower Peninusla would be split in half between Ohio and Indiana.
Anybody want to throw out names for the Lower Peninsula (and Upper if you want) should the 3 state plan had succeeded.