Does the Bible say that a daughter born of a Jewish mother is automatically Jewish herself?
The Bible never specifically says that a child of a Jewish mother, whether natural born or converted, is automatically Jewish. This has been a longstanding tradition among Jews, and has been confirmed by rabbinic court. It has been a sensible tradition, since the only sure parentage, before DNA comparison, is that of the mother. It also allows children of rape to be confirmed in their ethnicity without dispute.
There is biblical example of this principle, even if it is not specifically stated as law. “Then came he to Derbe and Lystra: and, behold, a certain disciple was there, named Timotheus, the son of a certain woman, which was a Jewess, and believed; but his father was a Greek: Which was well reported of by the brethren that were at Lystra and Iconium. Him would Paul have to go forth with him; and took and circumcised him because of the Jews which were in those quarters: for they knew all that his father was a Greek.” (Acts 16:1-3) Paul stood in for Timothy’s father in the bris (circumcision), and sometimes referred to Timothy as his son, as a result. This was necessary because Timothy would be considered Jewish because his mother was Jewish.