Coroner might be fun. Being Sheriff in a corrupt little Southern county could be a hoot as well.
Lord Chief Baron of the Exchequer. It just sounds cool.
Surveyor.
Vice President of the United States of America.
I’d work my hardest to get that job back to what it was before Cheney: sleep in, give the President a quick call to see if he’s died in office; if not, roll over and catch another couple of winks. Big mug of coffee with the paper, then maybe wander down to the Senate and see if they need a tie-breaking vote. If not, time for a nap. Make sure black outfit is clean and pressed for the next foreign funeral.
I’d be serving my country with honor and well-rested. What’s not to like?
Except that you’d have to spend your entire budget on new police cars, since they seem to crash a lot 
None. I would be a disastrous politician.
Wait a second. The Pope is elected, right? I mean, you didn’t say “elected by the people.”
This. Perks of being recognized and people always taking you out to meals, being invited to cool, exclusive things…yet none of the headaches of making real decisions in Washington for years in relative anonymity.
The NSW Legislative Council would be pretty cushy, I think. Eight year terms, generous pension arrangements, state-wide vote, so no specific group of constituents. Nothing much really to do either, if you were a backbencher for one of the major parties.
Are they normally elected?
President or a Supreme Court Judge. I don’t think I could screw up the job any worse than Bush. As for the SCJ, it’s a life long job, and it sounds interesting.
Potus.
I could go for President as long as I didn’t have to run for office. I would certainly do a better job than the current chimp.
Pope – from which position I would be an exemplar of debauchery, generally degrading the position of the Church in the world.
They’re appointed by whatever the modern version of the Colonial Office is; but the appointment is, I believe, made by a board of Important People. Hence, there’s a sort of election happening there and it fits within the OP’s requirements. 
Chairman of the dominant party in a fairly-large county.
Limited public oversight, no running for office myself, the ability to select those that do run, the chance for personal enrichment through control of some essential business, and the ability to have influence without responsibility.
I can’t think of anything better.
I’d probably take POTUS. The job really isn’t that hard. You have access to expert advice on any subject, and you decide between the alternatives presented. 4 years, maybe 8, and you retire with a pension and write your memoirs.
I’d actually rather be a judge…probably in a trial court, but an appellate gig would work.
Federal Judge would be awesome, but is an appointed rather than elected gig.
Not really an elected office, but Chairman of the Federal Reserve. A job I actually could do.