Why did the RLDS start giving the priesthood to women? There is a common belief amongst mainstream LDS that the reason the Community of Christ (formerly the Reorganized Church of Jesus Christ of Latter-day Saints) started ordaining women with the priesthood is because they ran out of male descendants from Joseph Smith to lead the church. 

What's the real reason that the RLDS started ordaining women?
Whatever the RLDS Church's actual motivations, the 1984 decision to ordain women didn't exactly come out of nowhere, according to the Encyclopedia of Mormonism's entry on the RLDS Church:

Section 156:9-10 meant that the church would now move ahead with women's ordinations, a breakthrough foreshadowed by events dating back to 1970. Local pastors had been initiating priesthood calls for women since 1974, but no clear precedent permitted actual ordination. Now, the conference's approval of section 156 created the context for the ordination of women, the first ones being ordained November 17, 1985.

Despite this, the decision was still quite controversial, and precipitated many churches breaking away from the RLDS Church.
William D. Russell, a historian and an RLDS member, makes a case in his article &quot;Grant McMurray and the Succession Crisis in the Community of Christ&quot; that the decision had nothing to do with running out of male descendants of Joseph Smith:

President [Wallace] Smith introduced the document that became Section 156 of the Doctrine and Covenants, approving the ordination of women and also announcing plans to build the temple. After a long debate, delegates approved it by a vote of approximately 80-20 percent. Some thought Wallace B. Smith—having three daughters and no sons—had introduced women's ordination to allow one of his daughters to succeed him. But since Wallace ultimately recommended a male successor outside the family [W. Grant McMurray], it appears that Section 156 was not based on this motivation. Although Wallace has never, to my knowledge, publicly discussed the background to this revelation, he clearly believed the male-only priesthood was based on culture and tradition, not divine will.

Here is the relevant portion of D&amp;C 156:

The following is also presented as the voice of the Spirit:
...
I have heard the prayers of many, including my servant the prophet, as they have sought to know my will in regard to the question of who shall be called to share the burdens and responsibilities of priesthood in my church. I say to you now, as I have said in the past, that all are called according to the gifts which have been given them. This applies to priesthood as well as to any other aspects of the work. Therefore, do not wonder that some women of the church are being called to priesthood responsibilities. This is in harmony with my will and where these calls are made known to my servants, they may be processed according to administrative procedures and provisions of the law.