Assuming the will is not updated, will Bernice still inherit their
share of Alice's estate?
What does Bernice have to do, if anything, to show they were
Prove to the satisfaction of the executor (who probably knows Bernice personally), and if necessary because of a dispute between the executor and Bernice, to a court with probate jurisdiction, that the person that Alice intended to leave the inheritance to in the Will at the time it was drafted was the person who is now known as Bernice.
Usually, this is straightforward.
For example, a common fact pattern if for a will to set forth a statement of family naming the children of the testator and if better drafted their dates of birth, and then using the names set forth in the statement of family to describe the amounts devised to each child. In that case, if Bernice can establish that Bernice is a child of Alice with the birthdate set forth in the statement of family in the will who is called Bob in the Will, Bernice will almost certainly prevail and receive the inheritance.
The task can be more difficult if the person named to receive the inheritance is not a family member (e.g. a neighbor or friend), has a common name, and is not described further in the will, although it is still usually obvious who was intended, with testimony about the relationship of Bernice to Alice at the time the will was drafted, about the fact that Bernice was called Bob by Alice at the time the will was drafted, and about the absence of any other people whom Alice called Bob to whom Alice could plausible have had any donative intent at the time the will was drafted.
If there is any risk of confusion, a better practice is to include additional detail about such a person, such as describing where that person lives or what their relationship to the testator is in the will itself. Using middle names or stating relationship to a third person (e.g. "Bob William Green, the son of my next door neighbor, Martha Evilton") can help avoid later ambiguity regarding who the testator intended to be the beneficiary.
Another way that ambiguity can arise is in cases where someone's family has followed the baneful practice of giving fathers and sons the same name, appended with a Sr., Jr., or ordinal number (e.g. the third) and no suffix is included in the description of who is to receive the gift described in the will. A lawyer who does this in a lawyer prepared will has probably committed malpractice, although there won't always be someone with standing to sue on that claim.