Last year I read a book written by a friend of JFK Jr., who speculated that he did consider making the race for the NY Senate seat in 2000, but his marital problems at the time (Carolyn having difficulty adjusting to the limelight) gave him second thought. John Jr. was annoyed with Hillary for not giving him advance notice that she was running, though.
The author suggested that JFK Jr. was thinking about running for the Senate from New Jersey in '02 or '04. Jackie O had owned a cottage on the Jersey shore that John Jr. inherited, and could have claimed as a residence.