That's an interesting possibility indeed. I like Strickland, and I think his kind of more blue-collar, Midwestern Democratic views have been put on the back burner at the White House. I wouldn't mind seeing him go there at all (though there would be a chance he would be marginalized there fairly quickly, like many others who might have brought such views to the administration, including Hillary Clinton, whose profile has been very low for a while now).
We just passed the midterm elections. This is the time when many Cabinet secretaries (regardless of how the president's party did in the midterms) start looking for the exits. The jobs are brutal, underpaid, and often thankless, though they can be builders of prestige and careers. I don't know if Obama would consider Strickland for a Cabinet appointment and I don't know what Cabinet positions may come open in the next three months or so. Nevertheless, if something of that order materialized, I'd wish Strickland all the best in D.C.