That is a great question but I am afraid I couldn't find an answer for you. Some of the first ladies didn't order their own pattern and Barbara Bush may be one that didn't. I was in the dish room at the White House during her son's administration and I don't remember seeing a display of her china.
My favorite pattern is Hillary Clinton's china followed by Mamie Eisenhower's and Nancy Reagan's dishes. Of the older first ladies, I like Sarah Polk's pattern best. She was in the White House from 1845 to 1849.
The patterns I like least are Michelle Obama's, Rosalynn Carter's and Harriet Lane's. Harriet served as first lady for her uncle, President Buchanan. You can see a picture of her pattern in an article I wrote at https://owlcation.com/humanities/Harriet-Lane