I tussled with this question for a few years because I see the logic of responses like Anne's: if we believe in a church led by prophets who really are guided by God, who are we to pick and choose which policies we agree with? Surely we can't sustain the Quorum of the Twelve with our right arm while seeking to undermine or contradict them with our left?
But most of my own spiritual development recently has hinged on a conviction I've come to hold about the nature of revelation and spiritual autonomy. While I believe (most of the time - the trajectory of my faith might be best described as "sinusoidal") that Church leaders are inspired by God, I also believe that they get things wrong. I think they got polygamy wrong (which, I feel, guppy demonstrates convincingly in Board Question #91105). I think they got the ban on blacks holding the priesthood wrong. I think they've got the anti-homosexuality crusade wrong. The scriptures teach us that apostles and prophets get things wrong all the time, that personal failings, lack of faith, or the sheer weight of blinding cultural traditions often lead them to do things that aren't in line with the will of God. Take Jonah, for instance, whose self-righteousness got in the way of him going humbly to preach to the people of Ninevah; or Peter, whose not-unjustified fear of the frenzied political climate drove him to deny Christ three times; or, perhaps most applicably, Paul, who was pretty convinced that women were second-class citizens in the Church (see 1 Corinthians 11:5-9 and 14:34-35) and who affirmed the institution of slavery (see Philemon).
So what seems to me a given—that leaders are fallible, that they make mistakes, that the things they say or the policies they implement are not always inspired by God—can only mean one thing: we must supplement top-down revelation with personal revelation. That's why we have personal revelation. It's crucial to both our own eternal development and the trajectory of the restored Church that we exercise spiritual autonomy in seeking answers for ourselves. We must ask for personal spiritual advice on which parts of our leaders' teachings are true.
And we must do so not just in the false, constrained way we often do by preemptively assuming what the right answer is. My experience has been that the Church often encourages us to ask for spiritual confirmation of prophetic counsel but also insists that if we don't receive it, we're the ones in the wrong. That's like holding an election but only putting one candidate on the ballot; it's fundamentally coercive, and at its heart, it's little more than a sham. We are not truly seeking revelation if we don't at least entertain the possibility of error, if we aren't at least prepared for God to whisper, as He may occasionally do, "No, actually, they've got that wrong, and you have to help me change their minds." Is it so inconceivable that there might be a third line of communication, that a God who is patient with the failings of those he calls and who abhors coercion might start the rumblings of change in the hearts of the humble, in the hope that their leaders will listen?
I suspect that the Church as an organization is at least as much human as it is divine. The New Testament is extraordinarily instructive in this regard: Jesus bursts onto the scene and spends three short years teaching a radical, sublime new religious philosophy and then, poof, He's gone. The apostles are left with their minds gloriously opened and their hearts forever changed, but with no idea what to do next. They're tasked with turning a ragtag bunch of followers into an organized religion with a unified structure and set of canonized doctrines, because although the Savior introduced all of the core tenets of the faith while He was with them, He's left them to sort out a lot of the details on their own. So what happens? The new church claws its way forward through clashes and missteps and disagreements and feedback. For example: a lot of ink has been spilled over how to reconcile James 2 ("faith without works is dead") with Ephesians 2 ("for by faith ye are saved...not by works"). I'm no theologian, and I certainly don't want to dismiss a complex and meaningful doctrinal discussion with too easy a wave of the hand, but the most obvious explanation to me is that James simply disagreed with Paul about how to interpret Jesus' teachings on salvation, and both wrote essays about it that have been enshrined in scripture. In another example, this one from Galatians, Paul recounts a literal, heated confrontation between himself and Peter over the issue of circumcision after Christ. The New Testament after the Gospel of John is basically a record of Peter and Paul and others bumbling around trying to develop a cohesive idea of what "Christianity" meant—sometimes based on clear, direct revelation, but mostly, it seems, just going with their guts.
Is the Church today really so different? Like the organization the ancient apostles were trying to build, the modern restored Church is directed by God but mediated by humans who are, well, all too human.
All of this is to say that I think the Church should be much more tolerant of dissenting viewpoints from its rank-and-file members. Other organizations improve and grow by the clash of ideas; forms of human reasoning like critique, philosophy, and dialectics can help us tackle difficult problems and keep improving the solutions. One of the severest deficiencies in Mormon education, in my opinion, is the lack of a thorough schooling in Christian philosophical history, in great thinkers like Augustine, Aquinas, Luther, Erasmus, More, and Kierkegaard. If we were to study them thoroughly, we might have a better understanding of just how huge a debt our theology owes to theirs—which might in turn give us a sense of the extent to which our Church was shaped by centuries of reasoned, human debate even before it tumbled into being in the spring of 1830. The Church resists any method of change that relies on dissent in an egalitarian forum because, as an organization that claims to be led by God, it (rightly) doesn't want to be subject to the caprice of contemporary politics or led astray by false prophets. But in trying to insulate itself from criticism, it has forgotten just how human many of its key elements are, and so has insulated itself from significant forms of human improvement.
There's an interesting strain of Mormon thought that doesn't get enough emphasis in standard Church lessons about choice and accountability. It has to do with consent, with the notion that part of being free agents entails being able to bestow or withhold permission for certain things to happen. Its most emblematic form is baked into the structure of the Church, and it happens every time someone gets a new calling: we raise our right hands to give them our sustaining vote. The word "vote" here is no accident, as it marks a formal expression of our will in the matter of ecclesiastical governance, and it's a crucial part of exercising our agency. And as we know from studying the plan of salvation, learning to righteously exercise our agency is fundamental to becoming what God intends us to become. This blurb from the website By Common Consent expresses it beautifully:
Since its inception, Mormonism has embraced the principle of having each member of the church exercise free will. Members are invited to individually study things out in their minds and verify whether the teachings of the church are the will of God. When the leadership of the church present decisions, the appropriate body of members is asked to sustain or oppose the action, discerning God’s will through personal revelation. Thus the revelation given to Joseph Smith in July 1830: “…all things shall be done by common consent in the church, by much prayer and faith, for all things you shall receive by faith” (D&C 26:2). Our site celebrates this holy interplay of personal agency and revealed will.
But in the contemporary Church, the act of sustaining has become merely a form, an empty signifier. Raising your hand when the second counselor, just to preserve the ritual, not in a million years expecting anyone to respond, asks "Any opposed?"—that is, actually, sincerely dissenting—doesn't really seem like an option. And in such a world, we're back to the ballot that lists only one candidate and the phony election that represents a triumph of coercion over consent. When honest disagreement becomes impossible, that's the end of spiritual autonomy, the sacred independence that is the whole point of this life and the next. Our development as spiritual beings depends on aligning our will not with the Church but with God, and how will we ever reach the sort of maturity of judgment required to be gods ourselves if we can't exercise enough spiritual autonomy to distinguish between the two?
That was an overlong preface for my answer to your real question, which is yes, I think God intends women to have the priesthood. Why? Because I've prayed, with as open a heart as I can muster, to ask whether it is His will that the priesthood be restricted to men. And the answer I've felt is, "No, actually, they've got that wrong, and you have to help me change their minds."
P.S. Above, Professor Kirke linked to a blog post that I completely disagree with (It's short; go read it so that you know what I'm reacting to). I want to respond to a couple of points. First, desiring something is not the same thing as coveting it. We all have righteous desires that we hope God will fulfill in this lifetime, but we wouldn't call it "coveting" to hope that God will bless us with the strength to tackle single motherhood or the intellectual skill to pursue a graduate degree or the opportunity to move to a place where a temple is within an hour's drive. Why is it righteous for a young man to sincerely desire to be ordained to the Aaronic priesthood but unrighteous for a young woman to desire it? Second, if the possession of the priesthood has nothing to do with the distinction between male and female but rather with whom God calls and whom he doesn't, do we think it's likely that he's "called" literally every righteous man over the age of twelve and literally no women? I suspect that more often than not, calling is linked with desire—that is, God calls those who desire to serve him and qualifies them by giving them the tools they need to do so. I have a lot more to say about this but there goes the end of Alumni Week, and with it all my writing privileges.