The first thing that came to mind was of the conceiving of Christ by Mary. In this aspect, the concept of the Heavenly Mother coming to Mary for her to act as a surrogate is really beautiful. ...as opposed to the creepy Brigham Young quote: "if the Son was begotten by the Holy Ghost, it would be very dangerous to baptize and confirm females, and give the Holy Ghost to them, lest he should beget children, to be palmed upon the Elders by the people, bringing the Elders into great difficulties." (JD1) Oh, Brigham Young. Anyway, there's some weird sexual tension in the immaculate conception, and this would fix that for me, with one mother passing responsibility to another.
The second issue is that in LDS doctrine, the Holy Ghost is disitinct from Heavenly Father and Jesus Christ in that (s)he doesn't have a body. From lds.org:
The Holy Ghost is the third member of the Godhead. He is a personage of spirit, without a body of flesh and bones. He is often referred to as the Spirit, the Holy Spirit, the Spirit of God, the Spirit of the Lord, or the Comforter.We obtain the doctrine of embodiment from the first vision, where Joseph Smith sees God the Father and Jesus Christ. This is backed up by the fact that the resurrected Christ is touchable in both the New Testament and The Book of Mormon. Much later, Joseph Smith gives instruction on the nature of the Godhead, which is recorded in D&C 130:22:
The Father has a body of flesh and bones as tangible as man’s; the Son also; but the Holy Ghost has not a body of flesh and bones, but is a personage of Spirit. Were it not so, the Holy Ghost could not dwell in us.
The nature of D&C130, however, seems to be more conjecture than anything else--prophets are entitled hypothesize too. JS backs up his assertion with logic (as did BY in his above quote), which to me indicates that it is possible he came to that conclusion on his own (well, it's possible he came to everything on his own, but that's another story). Of the LDS Godhead doctrine, I find the Holy Ghost's required disembodiment to be the least convincing. Still, however, it would be a major obstacle in any sort of official consideration of the Heavenly Mother-Holy Ghost theory. (Hey, that sounds kinda like the Adam-God theory. Maybe we just need a wacky prophet in order to swing it. Packer, I'm counting on you.)
The third issue I want to consider here is the role of women in general: what would Heavenly Mother as the Holy Ghost imply for the rest of us? There are sufficient reasons for not wanting to propagate a doctrine of Heavenly Mother at all, namely that without a divine role model, women have more choices; as long as society is patriarchal, having an archetype may not be the best for women. If the Holy Ghost were the female archetype, though, would that imply that women are inherently comforters, that they bring wisdom and peace? Would we dispense with the fire metaphor as being too masculine? Given that we know little about the Holy Ghost, would it take on an even more obscure and less understood role? Would we impose feminine characteristics on the Holy Ghost? Would doctrine such as this humble the patriarchy enough to enable true equality between men and women? Or would it encourage further pedestaling of women instead of giving them any real say?