They never reuse a clue exactly, but there are certain topics that are essentially cannonical - US Presidents/VPs in order and by time frame, for instance. Academy Awards. European Monarchs. The wives of Henry VIII. Shakespeare's works.
WH Harrison is the only shortest-serving POTUS and the only "first to die in office," but there are many ways to ask about these singular facts - who succeeded him, what year did he die, how long was his tenure, what killed him, what was his nickname, etc. By extension there are a number of ways to ask related questions about his successor Z. Taylor.
In many ways, a Jeopardy clue is like a word problem in math. You're not so much "answering a question" as much as "seeking keywords." Just this week - Category: "Vice Precedents" Clue: "He was the first vice president to assume the presidency upon the death of a president."
This is a WH Harrison/Zachary Taylor question.
If you search the J Archive for Zachary Taylor there a pile of ways to ask for his name:
He was sworn in as vice president March 4, 1841 & as president April 6, 1841
As the U.S. Constitution was vague, this V.P. decided to name himself President instead of calling for an election
1841 pair whose nicknames included "Tippecanoe" & "His Accidency"
If you can connect the dots among the keywords quickly enough you'll get to the right answer.
Sometimes they do barely change the questions, though...for instance, there are two almost identically worded straight up Harrison/Taylor questions...
This week's - above -
"He was the first vice president to assume the presidency upon the death of a president."
And 9/26/06 -
"He was the first vice president to become president upon the death of a president"