It's possible. And wouldn't that be something?
Under present Catholic Church canon law, you have to be ordained (e.g. a deacon, priest or a bishop) to become a cardinal. But it was not always so. The last non-ordained cardinal was Teodolfo Mertel, who died in 1899. But Pope Benedict XV in 1917 promulgated a canon law change that decreed that only priests or bishops could be appointed cardinals. Pope John XXIII then refined it by requiring that only bishops can be made cardinals, with special exceptions granted rarely for priests.
But canon law is not dogma, and just as it was changed to require cardinals first be bishops, it can be "unchanged." Theoretically, a non-ordained person could become a cardinal if the pope desired a change in the law. But what about a non-ordained is the person is a woman?
As far as I could find out, there is no prohibition in canon law against a woman becoming a cardinal (or even a pope--remember Pope Joan?*) If so, there is no legal impediment against women cardinals.
But tradition is a powerful force in the Catholic Church, so if Francis did appoint a woman cardinal, there would be powerful arguments over this precedent-setting action. (Again, I don't know if a woman ever has been appointed a cardinal in the church's 2,000-year-plus history).
Would Francis do it?
I don't know. I was having lunch with a priest friend who said we should expect some "surprises" from Francis. What surprises, I asked. He shrugged, as if to say, "If I knew then they wouldn't be surprises then."
Should Francis do it?
I'd love it. It would bring women into the highest levels of the church. The College of Cardinals exists to elect new popes, but some cardinals also serve as the pontiff's closest advisors. I don't think that it is beyond the realm of possibility that Francis would dip into the pool of many super-capable women in a church of more than 1 billion people to find such an advisor.
In a way, a woman cardinal could be viewed as something of a compromise between the church's dire need to elevate women into their rightful place in the church structure and the church's unmovable opposition to women priests. However, I wouldn't view it as a real compromise until women also can be ordained.
A woman in red, marching with the cardinals into the Sistine Chapel to elect the next pope? I'd love to see it. But why stop at one? Why not a bunch of them?
Pope Joan was a legendary female Pope who allegedly reigned for a few years some time during the Middle Ages. The story first appeared in 13th-century chronicles,  and was subsequently spread and embellished throughout Europe. It was widely believed for centuries, though modern religious scholars consider it fictitious, perhaps deriving from historicized folklore regarding Roman monuments or from anti-papal satire.
Francis' inaugural mass is today.
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