Agree with the prior answer, that this may be a mix-up.
There is one statement in your question that gives me pause - "He was married to my Mother at the time and it is very likely she was the beneficiary." Very likely is not going to be enough to get the stock transferred. You should ask for a copy of the beneficiary designation that was in effect at the time of his death. The executor of his estate may already have a copy, if there were other assets in the brokerage account.
Best of luck.
Sounds like a mixup to me. The transfer agent may not have ever recorded the stock in your mother's name. Your broker should be able to fix this for you.
The foregoing is not legal advice nor is it in any manner whatsoever meant to create or impute an attorney/client relationship.