While I'm a huge supporter of men and women being more equal, that theory rather rubs me the wrong way. I guess that's because it really sounds like the "Two Trees" theory, which I loathe with every fiber of my being, because it just provides a justification for the patriarchy and doesn't allow for any change.
Let me explain.
This theory was made by Valerie Hudson. Usually I adore Hudson - she's written some amazing things, including Sex and World Peace which is the most fantastic book EVER and everyone should read it. Basically Hudson writes that women need to have equal power as men and be on the same governing boards and governments as men because otherwise most of the problems in the world won't be solved. But then she goes and writes "The Two Trees" where she argues that everything is fine the way it is, and women shouldn't be arguing for a place at the table or for more representation in the Church. And this talk is all mental gymnastics on why women shouldn't be equally represented in the Lord's kingdom, though in other works she argues it's crucial women are represented in the governments across the world.
[Quick note: while basically all of this is drawn from the talk given at Fair Mormon which I linked to above, some of it comes from when she spoke at one of my classes. Furthermore, I'm not arguing for women to get the priesthood in this answer, I just agree with Sunday Night Banter's response below that we need to look for ways to give women a more equal role. Hudson is arguing for the status quo and effectively shutting down any woman or man who speaks up and says, "But wait, things aren't equal and should change."]
At first glance, the theory sounds pretty good. Just as there's two trees in the Garden of Eden, heaven and earth are seen as those two trees. Eve chose to partake of the tree of Knowledge of Good and Evil, thereby bringing life into the world. (Note: Eve partaking of the fruit was GOOD. I completely agree with Hudson here. Read her talk, read this, or ask me a question about it.) Hence, women's role in this life is as mothers. Not only do women bring life into the world, they also teach their children or others good from bad. Adam's tree, then, is the tree of life. Just as Eve brings life from heaven to earth, Adam uses his priesthood to bring life from earth to heaven. Both have their own ordinances: men have baptisms, the gift of the Holy Ghost, sealings, etc. Women have pregnancy and lactation (breastfeeding). While Eve may have been called to hearken unto Adam, "Adam hearkened first to Eve" by partaking of the fruit when she gave it to him. Men follow their mothers, get married, and "covenant to be the equal partner of your sweetheart." And we've misunderstood Genesis 3:16, because "'rule over' uses the Hebrew bet, which means ruling with, not ruling over." And here is the part that your question reminded me of: "Priesthood is a man’s apprenticeship to become a heavenly father, and I believe that women have their own apprenticeship to become like their heavenly mother."
That sounds really nice, doesn't it? But some cracks appear when you start to look closer.
First off, the main point of this theory is to say priesthood = motherhood. There are so many problems with this. Not all righteous women can be mothers, but all righteous men can be priesthood holders. Women must be married and be fertile to have children. Men do not have to be married and do not have to be fertile to hold the priesthood. Motherhood = biological and priesthood = worthiness. Hudson tries to avoid this by (at least in my class) pointing out that women also teach right from wrong, and "think of those poor orphans in Russia who had such struggles because they did not have a mother to teach them." What is she saying about single fathers? Can men not teach children right from wrong? Is a child raised by a single father as good as an orphan?
It's clear that women have no power over their tree without a man, but men have control over their tree without a woman.
Let's skip "Adam hearkened first to Eve" (we'll get back to that shortly) and get to men following women when they're children. So it's okay and justified for women to obey and hearken unto men because men followed their mothers when they were children? Would you be comfortable agreeing that all cultures and countries hearken unto and respect women because all children hearken unto their mothers? In addition, where do men "covenant to be the equal partner of your sweetheart"? There is no place in the Church that men covenant to that. (And Mormon married couples, remember what you agreed to.) The Family Proclamation blessedly says it, but also says men are to preside. If they are presiding over their wife, then they're not truly equal partners. I don't care what Genesis word is actually used, because the (mis)understanding we all have of it is engrained in our scriptures, our doctrine, our culture, and our temples. Christ told men to pluck out their eyes if they were tempted by a woman or what she was wearing, but we know from the multiple questions we've received lately that our culture does not care about that.
"Women have their own apprenticeship to become like their heavenly mother"? So I'm going to be unknown to most of my children? They're going to be told not to talk to me? They're never going to discuss me, or think about me, or wonder where I am or if I care about them? I'm not going to be able to talk to them? We paint our heaven in the similitude of what we see on earth, so is it any surprise that we see no woman in it, or if we do, that we refuse to talk to her or let others talk about her?
This is why I loathe this theory. Because it provides justification for the patriarchy. It tells men and women to be content with what we have now. It tells men and women to not worry about, to accept, the underrepresentation of women in Church leadership and heaven. It tells women to be content with her lot, ignoring that there is so much that is wrong that we need to talk about before we can change anything.
I don't want my Heavenly Mother's priestesshood because I don't know anything about her. All I've been told so far is that she gives birth. Her divine power is motherhood. Which is great, except when you realize the equivalent of motherhood is not the priesthood. The equivalent of motherhood is fatherhood.
Before I wrap up, I promised I'd address "Adam hearkened first to Eve." Hudson later expounded on this: "We know that for the endowment, for those of use en route to the Second Tree, the daughters of God hearken to the sons of God in their apprenticeship to Heavenly Father. I think it’s quite possible that en route to the First Tree there was also a covenanting, where the sons of God covenanted to hearken to the daughters of God in their apprenticeship to Heavenly Mother, and that Adam’s partaking of the fruit from the hand of his wife, Eve, was his fulfillment of that covenant." En route. En route. Hudson is saying that it's okay women covenant to obey men in this life because men covenanted to obey women in the preexistence. She is literally making up doctrine to justify women serving men.
This is what happens. Because there's no way to justify this. You have to literally make up doctrine in order to feel okay with women's position in the Church.
Sorry, that was probably quite a bit more than you were expecting. Basically (tl;dr) no, because the majority of Mormons view Heavenly Mother as a sacred baby-making machine, and giving birth is her priestesshood, and I don't (and I'm sure Heavenly Mother as well doesn't) want to be defined and limited by only that.
-guppy of doom