It seems to me that, at this point, there is only one course here that makes sense: Governor Paterson should appoint a well-respected figure who would agree not to run in 2010. (Judith Kaye, maybe?)
With a caretaker in place for the next two years, the other contenders could compete for the job the old-fashioned way–by going to the voters. By winning a Democratic primary (if she could), Caroline Kennedy would have a chance to prove her political bona fides. New Yorkers would learn what she stands for, and what she intends to do in the job. They would see how she performs in debates. She would be a stronger general-election candidate as a result.