Inspired by this thread, plus the fact that my grandfather actually did die on Election Day, 2008. I doubt that he voted (he had been mobile enough to physically make it to the polls until he had a heart attack a month earlier, and after that, I doubt that getting him an absentee ballot would have been particularly high on anyone’s priority list), but if he had, would it have counted? Does it matter whether the person dies before or after the polls open?
How would that even work? What if I voted in person and then died of a heart attack on my way out the door? How would they identify my ballot?
If the vote has been cast, then yes, of course it counts. It’s a validly cast vote, after all. If it has not yet been cast, then the right to vote does not pass on to the heir or next of kin.
Of course it counts, and as you note, there is not way to “uncount” it. Your ballot is secret, so your name isn’t on it.
What if the person dies while in the process of filling out the ballot?
True. I was thinking more about absentee-ballot situations, where (per the other thread) the vote apparently doesn’t count in some states if you die after mailing in your ballot but before Election Day. But you’re right that a vote cast in person has to be counted even if the voter dies on their way out of the polling place, so it stands to reason that an absentee vote would be equally valid.
Then no. It wasn’t submitted.
That came out in 2008. Barack Obama’s grandmother died a couple of days before election day. She had filled out an absentee ballot before she died. The election officials said it was a legal ballot because it was legally filled out before she died. The vote was duly counted.
No word on who she voted for. 
If the cities mayor dislikes the way the booth is voting, he declares it a crime scene and blocks the rest of the people from voting.
If he likes the way it votes, he suspends any such action under emergency powers and ensures the booth remains operating.
Even then, it only doesn’t count if they find out you died and match that to your ballot. That process will be imperfect at best.
I doubt that many mayors have the authority to declare a crime scene. In a lot of cities, including where I live, the mayor is a member of the city council and gets one vote, but is mostly a figurehead.
In Indiana, absentee ballots are sealed when cast, and not opened until election day, and counted at the poll where the voter lives (I have worked many elections). It is someone’s job to go through them and make sure that absentee ballots of deceased people get culled. I remember when a person my parents knew was very ill, and he voted absentee for someone whose campaign he had worked on many times in elections past. He died before the election, and his vote got culled.
Anyway, votes cast on election day at the polls are not signed. They are put into the voting machine and there is not way to cull the vote if the person votes early in the morning, and drops dead at 4pm.
Absentee ballots are not signed, but they are in signed envelopes, so they get to the right precincts. The clerks double check that everyone whose ballot they have is on the rolls, then they remove the ballots and lay them face down, and the envelopes are discarded. Then the ballots are turned over and counted.
So actually, if you vote absentee, and drop dead on election day, your vote still counts.
The rationale is so that sick people who may be not well mentally-- or at least weak and easy to influence, may be preyed upon by party workers trying to get as many votes for their candidate. It is legal to have help filling out the ballot if you need it, and people who are close to death are good targets for undue influence. If there is a good chance that it may be a wasted effort if they die before the election, the exploitation is discouraged.
It apparently did happen several decades ago, although to drunks more often than people in hospice. But the same thing still happened.
The inspector is in charge of the voting place, and has the authority to move the voting materials to a different area.
The mayor knows some guys who can MAKE it a crime scene. Capisce?
Jimmy Carter said in a few places in rural Georgia it was considered OK for a widow to vote 2 times , because they were also voting for their late husband. They stopped that when he was governor.