I've read that article previously as I follow Mental Floss on Facebook. I disagree with the article where it states "There are many theories on why
Bill became a nickname for
William; the most obvious is that it was part of the Middle Ages trend of letter swapping." I don't believe that
Bill began being used as a nickname for
William until the early-to-mid-nineteenth century. Before that, the only nickname was
Will. The article states that
William III was called "
King Billy" in the late seventeenth century, but although he is called
King Billy in Northern
Ireland and Scotland according to Wikipedia, it doesn't state that the term was contemporary to his time.
You just never hear of a
Bill before the early nineteenth century and I don't think it became really common until the mid nineteenth century. It was
Will Shakespeare, not
Bill Shakespeare.