The term “patriarchy” is controversial.
I used to be opposed to the term “patriarchy” and was settled on the term “complementarianism” to describe the relationship of the sexes. I was especially frustrated that many who called themselves “patriarchal” seemed to have weird “hyper-patriarchal” views. Perhaps that is still the case in some quarters. But a couple things have happened over the last several years that led me to reconsider the use of the term “patriarchy” and to embrace it.
First, the term “complementarian” has been greatly diluted. Many things that pass for “complementarian” are practically egalitarian (with perhaps a carve out for a male-only pastorate). This is certainly the view of, say, Tim Keller or Aimee Byrd. Complementarianism was a movement, in reaction to the rising feminism of a generation ago, but it is proving to be an inadequate one. Complementairanism has proved to be the "squishy" middle that is getting sorted out; more and more, Christians are moving towards either full blown egalitarianism (denying any significant differences between the sexes) and traditional patriarchy (in which the differences in the respective natures of men and women serve as the ground and basis for their differing roles in church, home, and society).
Second, during the “federal vision” controversy there was a great deal of emphasis put on recovering the language of the Bible itself. I came to realize that “patriarch” is biblical language (at least in most good English translations, and even if obscured in English, it is still there in the underlying Greek of the NT in numerous places). If the “federal vision” insisted on recovering the language of “baptismal regeneration” based on Titus 3:5 (“the washing of regeneration”), how much more should we be eager to recover the biblical language of “patriarch,” given that it occurs much more frequently in Scripture? The term "patriarch" occurs in the ESV translation of the NT six times; related Greek terms occur many more times, though the translation may hide that reality from English readers.
But we still have a key question answer: What does patriarchy mean? To give a more specific example, what does Peter mean in Acts 2:29 when he speaks about “the patriarch David”? The “arche” part of the word clearly means “rule,” not “source” (as some have suggested) in this context. The point is that David ruled Israel, and as a king he was a type and shadow of the Messiah. (In that section of Peter's sermon, the rule of Jesus is clearly the main theme.) Men are patriarchs in their households because they are the highest human authority in the home; each man is a David in his own household. This not to say that no one else has authority; there is a matriarchy corresponding to the patriarchy, but the authority of the wife/mother is not identical to that of her husband. Rather, just as the church (the bride of Christ) has real authority, but remains in subjection to Christ, so it is with the wife/mother in relation to the husband/father.
The term “patriarchy” can also be used to describe the bias towards male rule we find in Scripture and across cultures throughout history. It seems rather obvious that male rule is simply built into the structure of the world. This is not say there are not exceptions (e.g., Lydia, presumably a widow, was head of her household and there have been queens or female prime ministers who ruled their nations). But in general, male rule is an established and inevitable fact of life. Scripture underscores this in all kinds of ways (e.g., Isaiah 3:12). But while patriarchy is inevitable, it should be noted that this does not make patriarchy an unqualified good in a fallen world. There are good and evil patriarchs, good and evil patriarchies. The Christian faith represents one kind of patriarchal system, Islam another. Our Patriarch in Heaven is the ultimate model of a good patriarch, but Satan is a patriarch too (e.g., he is the father of the lying and murderous Pharisees in John 8). The patriarchy predates the fall; the fall did not destroy it, but it did distort it so that only by redeeming grace can the patriarchy function as God intended. The only patriarchy worth having and worth defending today is patriarchy in Christ. Christopher Dawson has done an excellent job showing hoe the gospel transformed patriarchy in the ancient world; a helpful summary can be found here.
Finally, some have wondered if the language of “patriarchy” somehow privileges the person of God the Father, and thus results in Trinitarian error. There certainly is a special connection between the divine Father and human fatherhood (Eph. 3:14-15). But in relation to creation, the entire Godhead, Father, and Son, and Spirit, is the Patriarchy, over and above the feminine creation. Father, Son, and Spirit are each masculine in Scripture. Each is given exclusively masculine pronouns. When the entire Trinity is spoken of, a masculine pronoun is used. While God can also act in feminine ways, this does not negate the fundamental reality: the Trinity is the ultimate and original Patriarchy; the Father is Lord of creation, the Son is Lord of creation, and the Holy Spirit is Lord of creation – and yet there are not three Lords, but one Lord.