I'm Doug, and I'm very sorry to hear of your mother's situation. My goal is to provide you with excellent service today.
I'm very sorry but if the cousin actually named a specific person to be the beneficiary of a bank account after death---also known as a Transfer on Death account, the assignment of a beneficiary takes precedence over anything which might have been in the will---because the transfer on death beneficiary is paid the money in the account outside of probate and like any other asset set out in a will, if the person who made the will disposes of that asset before death by selling, giving it away or by assigning it to someone else, the asset is deemed to no longer exist for purposes of the will.
Based on the facts you have provided, your mother has no legal right to the money that was in the bank account. I am so sorry,
I understand that you may be disappointed by the Answer you received, as it was not particularly favorable to your situation. Had I been able to provide an Answer which might have given you a successful legal outcome, it would have been my pleasure to do so.
If you have additional questions, you may of course reply back to me and I will be happy to continue to assist you further until your questions have been answered to your satisfaction.
Would you please take a moment to positively rate my service to you based on the understanding of the law I provided by clicking on the rating stars---three stars or more. It is that easy. That is the way I am compensated for having helped you.
Thank you in advance. I wish you the best in your future,