A:
A vote can be reconsidered later that same day or the next day (if any) of a session.
So, if your question is about a vote already taken, then reconsideration is too late.
Do your bylaws say that once rejected for membership, a person isn't eligible to re-apply?
If not, just take the vote again at another meeting (it can't be the same meeting).
If you can't renew the election question, you can rescind the previous black ball election by:
After the vote is rescinded, you can re-ballot.
Since the original method (black ball) was secret, I recommend a secret method be used in the vote to rescind, the easiest being by ballot.