* Queen Elizabeth I: She was a monarch and didn't have any children.
* Elizabeth Taylor: A famous actress, but her children did not pursue acting careers.
* Elizabeth Banks: An actress, and she has two adopted sons. It's unclear if they are pursuing acting careers.
* Elizabeth Hurley: An actress, and she has a son who is also an actor.
* Elizabeth Olsen: An actress, and her sisters Mary-Kate and Ashley Olsen were also child stars.
To get a proper answer, please specify which Elizabeth you're interested in.