The State Archives at the Minnesota Historical Society are a large, unwieldy collection, with some terrific online resources for family history buffs, including a name search feature to die for.
The site index gives a good, quick overview of what's available (but again...which topics are online vs offline is impossible to tell).
A good starting point for genealogy research is the Minnesota People Find search page where, in one fell swoop, you can search birth and death records, census records, veteran's gravesites, and an assortment of other databases. This is a fantastic tool, and one that other state archives would do well to emulate. Hats off to Minnesota!
The Archives' Visual Resources Database is surprisingly large, with over 100,000 digital images. Click the button for Show only those with digital images to restrict search results to online content.
A few other gems scattered about include:
Don't overlook the Minnesota Historical and Cultural Grants Program, that supports historical and genealogical work.
There are also a number of other sites with an assortment of Minnesota historical digital resources.