Regarding biblical lineage: Was there ever a time in the course of history when Hebrew lineage was recorded through the female?
Lineage was traced through the males throughout the Bible. The only exception is in the case of Jesus because his father was God. You can see this in all the lineages where the fathers are listed. Mothers appear only rarely in such lists.
Modern Jews claim the use of maternal descent based on "Nor shall you make marriages with them. You shall not give your daughter to their son, nor take their daughter for your son. For they will turn your sons away from following Me, to serve other gods; so the anger of the LORD will be aroused against you and destroy you suddenly" (Deuteronomy 7:3-4). Since verse 4 mentions turning the sons (in plural) away, they claim that it is referring to the descendants of the marriage and thus the argument that the mother determines the lineage.
Sadly, it stretches the actual statement. God told the Israelites not to intermarry with a select set of nations (Deuteronomy 7:1-2), not all other nations. The warning was against either taking husbands or wives from these nations because they would turn people away from God. "Sons" is being used in a generic sense to refer to both the men and women.
Another verse used is "Now the son of an Israelite woman, whose father was an Egyptian, went out among the children of Israel; and this Israelite woman's son and a man of Israel fought each other in the camp" (Leviticus 24:10). They then point out that her son was counted as an Israelite. What they ignore, however, are the cases where a descendant of Israel married non-Hebrew women and their children were counted as well. For example, Jacob, ancestor of the kings of Israel, had children by two non-Hebrew women. "And Judah saw there a daughter of a certain Canaanite whose name was Shua, and he married her and went in to her" (Genesis 38:2). His second wife, Tamar, also appears to be not of Hebrew descent. Joseph's wife was an Egyptian and his two sons Manasseh and Ephraim were the founders of two of the tribes of Israel. The usual way of avoiding the obvious is to claim that these men's wives all converted to Judaism. Yet, could that not be claimed for the woman whose husband was an Egyptian?
Yet another argument is that in "And Shechaniah the son of Jehiel, one of the sons of Elam, spoke up and said to Ezra, "We have trespassed against our God, and have taken pagan wives from the peoples of the land; yet now there is hope in Israel in spite of this. Now therefore, let us make a covenant with our God to put away all these wives and those who have been born to them, according to the advice of my master and of those who tremble at the commandment of our God; and let it be done according to the law" (Ezra 10:2-3). Much is made of the fact that only the men are mentioned as putting away their foreign wives and their children. The argument is that if the children were Israelites, they could not be put away. What is glossed over is that no child of certain nationalities could be counted as Israelites. "An Ammonite or Moabite shall not enter the assembly of the LORD; even to the tenth generation none of his descendants shall enter the assembly of the LORD forever" (Deuteronomy 23:3). Notice the "his," and understand that it is generic for male or female. If an Israelite woman married a Moabite man, their children would not be counted as Israelites either.
The maternal lineage is traced to the Mishnah, the codification of Jewish oral traditions. "The Mishnah (Kiddushin 3:12) states that to be a Jew, one must be either the child of a Jewish mother or a convert to Judaism" ["Matrilineality" in Wikipedia]. It was something that developed over the years but doesn't appear to have been prominent at least through the days of Jesus.