While the negatively charged rod is near the disk without touching it, a hand briefly touches the end of the post. Then the negatively charged rod is removed. What would happen to the vane?

that initially the weather vane was at rest, by this load that remained on the pole it would begin to move.

Explanation:

Let us carefully analyze the situation, when the bar is facing the index post a load of equal magnitude, but opposite sign on its surface, these two charges are in balance; When the hand touches the pole, it creates a path to the ground where the charges that were induced on the pole can be balanced with the charge coming from the ground, leaving a zero charge on the pole.

Now if the hand is removed, there can be no exchange of charges with the earth. When the bar is removed, the induced loads are redistributed in the post, but the excess loads that came from the earth that have the same value and are of a sign opposite to the induced ones remain, you want to sign that they are of the same sign as the charges of the bar.

In summary, after the process, the post has a load of equal magnitude and sign (negative) that of the bar.

If we assume that initially the weather vane was at rest, by this load that remained on the pole it would begin to move.