Final answer:
The shared name of the only pair of presidents who weren't related is Roosevelt.
Step-by-step explanation:
The shared name of the only pair of presidents who weren't related from the five sets of presidents with shared last names is Roosevelt.
In the set of presidents with the last name Roosevelt, there were two presidents: Franklin D. Roosevelt and Theodore Roosevelt. They were distant cousins, but not closely related.
Therefore, the shared name of the only pair who weren't related is Roosevelt.