The idea is not specifically mentioned in the Bible. However, there is a biblical concept of "headship" that applies to every husband, which would seem to apply both to a woman's father prior to her marriage.
For the husband is the head of the wife, as Christ also is the head of the church, He Himself being the Savior of the body. Ephesians 5:23 NASB
The root of this comes from Genesis 2, when God brought the woman to the man--and not the other way around.
The Lord God fashioned into a woman the rib which He had taken from the man, and brought her to the man. Genesis 2:22 NASB
This is illustrated further in Genesis 3, where the woman sins first, but God's interrogation begins with Adam.
When the woman saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was desirable to make one wise, she took from its fruit and ate; and she gave also to her husband with her, and he ate. Genesis 3:6 NASB
Then the Lord God called to the man, and said to him, “Where are you?” 10 He said, “I heard the sound of You in the garden, and I was afraid because I was naked; so I hid myself.” 11 And He said, “Who told you that you were naked? Have you eaten from the tree of which I commanded you not to eat?” 12 The man said, “The woman whom You gave to be with me, she gave me from the tree, and I ate.” 13 Then the Lord God said to the woman, “What is this you have done?” And the woman said, “The serpent deceived me, and I ate.” Genesis 3:9-13 NASB
The idea of "headship" is certainly a controversial topic today. However, it is likely the concept of which the modern marriage ceremony is built. The giving of the bride by the father to the husband symbolizes the transfer of responsibility.