In 1832 Samuel Bryan of Marion County Indiana applied for a Revolutionary War pension. He had relocated from North Carolina to Indiana late in life. His wife applied for a widow's pension after his death. Included in their pension application was a document supplied by their son Luke which contained the following:
[Luke Bryan submitted the following with his mother’s application for a pension, and he deposed that it was written by his father, Samuel Bryan.]Samuel's great grandfather was not named in the application. Nor was the name of his grandfather or father. This would change and by the time this genealogy was found in print Samuel's ancestors had aquired some names.
My great grandfather Bryan was a Dane born in Denmark & rais’d in that Kingdom where he married a wife & lived untill he had a sone born whome he called Morgan after which he remov’d to Ireland where he lived untill said Morgan came to manhood who left his father in Ireland & came to Pensylvania in Amerricia where he Married a woman by the name of Martha Strode the daughter of a man by the name of Strode a Hollander who had moved to France where he resided with his wife untill he had three children, he & his wife being protestants, in time of a great persecution fled for their lives, bound for Pensylvania in Amerricia but himself & wife sickened on the seas & died before they arrived to the end of their voige....
William Smith Bryan is an interesting character. He is first written about in a book published in 1876 called The Pioneer Families of Missouri, written by William Smith Bryan, his descendant.  Mr. Bryan wrote that his ancestor landed in Virginia by way of Ireland in 1615, saying he "arouse the hostility of the British Government by a too ardent Irish patriotism and was deported as a rebellious subject." He also claims that William Smith Bryan was the only living lineal descendant of Brian Boru, a high king of Ireland from the 10th century. Mr. Bryan goes on to say that William Smith Bryan had eleven children but the name of only one was known; Francis, who returned to Ireland to reclaim the family lands.
This Francis was unsuccessful in his attempt to regain the old family land and for whatever reason is said to have fled to Denmark. In Denmark two sons were born; Morgan and William. Morgan inexplicably became the Standard Bearer for William of Orange and was present at the battle of the Boyne in 1690. He then left Europe and immigrated to Pennsylvania in 1695, where he married Martha Stroud. 
On page 132 of the Pioneers of Missouri, a second William Bryan is introduced. He is said to have come from Wales with Lord Baltimore in 1650 and settled in Maryland. His wife was Irish and they had three children; William, Morgan and Daniel. A descendant of this William, another William, settled in Roan County, NC and married a woman named Sally Bringer. The had eleven children, one of whom, a daughter named Rebecca, married Daniel Boone. 
By 1915 the story had changed quite a bit. In the book The Shearer and Akers Family, by J. W. Shearer, William Smith ? Bryan, his question mark, was a descendant of Sir Francis Bryan. No wife is identified. William Smith Bryan is said to be a landowner in Ireland, probably County Clare. In 1650 he and his family were deported by Cromwell for being a rebellious subject. Accompanying him were eleven sons, but only two are given in this book, Morgan and Francis.  This newer version sets the action forward by 35 years into the Cromwell era.
Again, Francis is said to have returned to Ireland to reclaim the family land but was persecuted by the English and fled to Denmark. Morgan is identified only as a possible son and is said to have been in Norfolk, Virginia in 1663. His son Morgan married a woman named Martha Stroud and ended up in Davie County, North Carolina. 
In a 1917 publication William is described as an Irish land owner with eleven children who was deported by Cromwell's forces in 1650 for being a rebellious subject. Francis returned to Ireland in 1677 and fled for Denmark where his sons William and Morgan were born. William married a Margaret and they lived in Ballyrooney, County Down, Ireland. William and Margaret and their son John left Ireland after John was arrested for poaching. 
In 1922 a book called Notable Southern Families was published.  William Smith Bryan is also discussed in this book, in which he is said to be the son of Sir Francis Bryan. He was deported in 1650 for being an undesirable citizen. He arrived in Virginia in 1615 with his family and a boatload of household goods. This book makes the amazing claim that William Smith Bryan was the ancestor of the O'Briens who were the Lords of Inchquin. The book repeats the story of Francis' return to England and the troubles that led him to Denmark, his marriage to Sarah Brinker. This version of the story says that Francis eventually returned to live in Ireland. It also supposes that Morgan Bryan was a son of William Smith Bryan. Morgan, this time, left Ireland for Pennsylvania. 
Another book published in 1922 was a history of the Boone Family. This book gives two versions of the story of Morgan Bryan. In the first, Morgan grew to manhood in Ireland and then left for America, settling in Pennsylvania where he married Martha Stroud.  The second version is the William Smith deported version but has Francis returning for his land in 1650. Again Francis goes to Denmark where son Morgan is born. Morgan of the battle of the Boyne comes to Pennsylvania in 1695. Francis died in Belfast in 1694. The author says that he does not know which if either were the correct story.
A 1962 article in the Virginia Magazine perpetuates the story of Francis Bryan returning to Ireland to reclaim the family estates. His son Morgan, then living in poverty, sailed, possible under indenture to Pennsylvania. 
In 1965, in the National Genealogical Society Quarterly, Volume 53, it is Francis that is the standard bearer for William of Orange in 1690.
Skip forward a few decades to the age of the internet. In 2011 this was posted on a ancestry.com message board:
Sir William Smith BRYAN, 8th Great-Grandfather. PMC" Prince William of Ireland", Deported in 1650, to Gloucester Beach, Virginia as a "Rebellious Subject." Marriage 1: Countess Of Ormond Catherine MORGAN, b: 1594 in , Claire, Ireland.
Married: 1620, in , Claire, Ireland.
Note: In 1650, William Smith Bryan, the Grandson of Sir Francis Bryan, declared himself Heir-to-the-Throne Of Ireland, and fought against Cromwell, from the back of a White horse. Defeated by sheer numbers of the Puritan army, Bryan was deported to the Colony of Virginia in America, together with "twenty-one sons and grandsons." Declares himself Heir to the Throne of Ireland.So in about 150 years William Smith Bryan has gone from a Danish man to 1615 Irish rebel to a full on pretender to the throne, a knight no less, seated on a white horse, battling the forces of evil Cromwell. He is now married to the Countess of Ormand! And, he has accrued 21 sons, but still only two that can be identified.
William Smith O'Brien
So where did this William Smith Bryan stuff come from and is any of it real, or is William a mythical fantasy ancestor. My vote is that he is an imaginary character. I believe that he is based on a real life man named William Smith O'Brien. William was the younger son of Sir Edward O'Brien and his wife Charlotte Smith, daughter of William Smith. Sir Edward was also Baron Inchquin. William Smith, despite being a member of the upper class and a protestant took the side of the poor roman catholic Irish population. He encouraged the use of the Irish language and sought relief during the famine. In 1848 he was charged with sedition as the result of a ill executed 'rebellion.' Sentenced to death, the country took up a petition to spare his life. He was deported to an island off of Tasmania in exile. Eventually he was release, spent some time in Brussels before returning to Ireland.
Doesn't that sound familiar? A son of a Lord, Irish rebel, deported, spent time in a European country before he eventually returning home to Ireland. William Smith Bryan and William Smith O'Brien. I believe that the life of O'Brien was transferred to a unknown ancestor by a Bryan biographer in the past. This fable has been added to over the years, resulting our knight in shining armour, riding a white horse.
 William Smith Bryan, Pioneer Families of Missouri, (St. Louis, MO : Bryan Brand & Co., 1876) viii; digital images, Archive (https://archive.org/stream/historyofpioneer00bryauoft#page/viii/mode/2up : accessed 2 September 2016).
 Bryan, Pioneer Families, viv.
 Bryan, Pioneer Families, 132.
 J. W. Shearer, The Shearer Akers Family, (Sommerville, N.J : Press of the Somerset Register, 1915) 11; digital images, Archive (https://archive.org/details/shearerakersfami00shea : accessed 2 September 2016).
 J. W. Shearer, The Shearer Akers Family, 11.
 George Norbury Mackenzie, Colonial Families of the United States, (Grafton Press, 1917) digital images, Google Books.
 Zella Armstrong, Notable Southern Families, (Chattanooga, Tennessee; The Lookout Publishing Co., 1922) 33, digital images, Google Books, (https://www.books.google.com : accessed 2 September 2016).
 Zella Armstrong, Notable Southern Families, 33.
 Jesse Procter Crump, The Boone family: a genealogical history of the descendants of George and Mary Boone, who came to America in 1717 : containing many unpublished bits of early Kentucky history : also a biographical sketch of Daniel Boone, the pioneer, by one of his descendants
(Buffalo, New York: Tuttle Co., 1922) 505-506, digital images, Google Books, (https://www.books.google.com).
 Charles W. Bryan, "Morgan Bryan; Pioneer of the Opequon and Yadkin," The Virginia Magazine of History and Biography, Vol. 70. no. 2 (April 1962) 154-164, digital images, JSTOR ( http://www.jstor.org/stable/4246837 : accessed 2 September 2016).