Although Judge William B. Bush (born ca 1796), the son of Dr. John H. Bush and Mary Rowland always appeared named as William B., I have never found any hint as to what his middle initial stood for. I really can't even give you a clue. He did name his own son William Hamilton Bush and he was sometimes referred to as "Ham". Have you any clues?? William Hamilton Bush's grandfather, Dr. John H. Bush was associated with a John C. Hamilton in Sumner County, Tennessee and the name Hamilton may have come from him.
As for Sally Bush Brown Brickey's son William B. Brown - his middle initial may have stood for Bush as he appears to have been named for his uncle - Judge William B. Bush. John K. Brown, Sally's first husband and the father of young William served with his brother in law, William B. Bush in the Missouri Mounted Rangers in 1814.