I would rather discuss flowers with women's names. It seems a lot safer than getting into this semi-political quagmire, but here goes. . .
When I first heard of putting a women's face on the $20 bill I thought it would be much like the presidents' quarters, one women for a while, then another, and another. Almost all of the women are reasonable, satisfactory choices for short time fame (and I'm talking the long list not just the final four).
However, my choice of the four would have to be Harriet Tubman. Probably the least appealing in photos -- which is something important to many in modern society. She was one amazing woman, totally selfless and courageous, returning again and again into slave territory to lead her people to freedom in the North against extreme odds and with a huge price on her head if captured.
Other women may be better known, but none can compete with the uneducated slave who was at the same time praised as "Moses" and cursed as "Satan" by the South.
I do rather like the idea of a Cherokee Chief replacing Jackson, however. I read "The Trail of Tears" to my students every year. It was a terrible time in American history with mistreatment of the Cherokee People (and the other 4 civilized tribes) an unpardonable crime.
Prior to 1861 we had Lady Liberty and Pocahontas, and since then men on the $20. Why choose? Let's just replace all US bills for the next 150 years with these great women since men got the last 150!