To properly answer your question there is additional information needed.
Is the stock is publicly issued or is it restricted, who is holding the stock at the present time, whose name is is the stock being held, is your mother the sole beneficial owner, how many shares and what is the market value of the stock.
If the stock is fully issued, it may be easier to just have the bank sell it.
If the stock is held in the bank's nominee name, then for your mother to sell it elsewhere, she will have to have the stock either transferred and shipped to another broker/dealer or have the stock moved to another broker/dealer through the ACATS system. Nonetheless she will have to open an account with another broker/dealer.
If the value of the stock being held is substantial, it may be easier, faster to have the bank sell the stock.
If you want to save commissions, then go through the gyration of having the stock moved to another B/D after you open account.
from the street