definitely stay
on the island. We used to go every year in the spring, and falling asleep at night listening to the clip-clop of the horses' hooves was wonderful. It is more expensive than staying off the island and doing a day trip or two, but you miss the real uniqueness of the island. My favorite times there were after the last boat of the day departed for the mainland!
We stayed in a couple of places, but this was one of my favorites:
http://www.tripadvisor.com/Hotel_Review-g42423-d271838-Reviews-The_Island_House-Mackinac_Island_Michigan.htmlIt's not in the main strip of town, but close enough that it's an easy walk. This was quite a few years ago, but it looks like the rating is still up there.
The Grand Hotel is cool, though I've never stayed there (too pricey). I checked a few dates in Sept, and it seems they're pretty booked up already, anyway.
This was another place we stayed once. It was very nice, but not close to town. They do offer carriages to town on a regular schedule, or it's a neat bike ride in (though don't recommend the night ride like we did once. It's downhill, and dark, and we almost wiped out several times, though we laughed all the way, in the blissful ignorance of youth. lol)
http://www.theinnatstonecliffe.com/As far as weather, Sept. is usually quite nice in Michigan, though like someone posted it can vary, but this is true even in the height of summer. It's just the way Michigan weather is.