The FitzSimons Family Name:
Spelling variations: Fitzsimons, FitzSimons and Fitzsimmons are the most common, but FitzSimon, Fitzsimonds, FitzSymons, Fitzsimmins, Fitzsymmons, Fitzsimon, Fitzsymons, FitzKimmons, FitzKimmins, Fitzkimmons, Fitzkimmins, and many more can be found. More information on spelling differences (particulary between England and Ireland) can be found here 
Preface: The prefix "Fitz" merely means "Son of". There is a misconception (pardon the pun) that it means "illegitimate offspring of" which might have arisen from the alleged bestowal of the name "Fitzroy" by royalty as a family name for the illegitimate children of their mistresses. However I digress.
Misspellings: See the history below, but more recently (especially from 1845 during the emigration from Ireland due to the great potato famine years) family names might be mis-spelled on records such as passenger lists. Remembering that literacy was not high among all classes back then, sometimes spelling was arbitarily written down by clerks who might apply their version of the spelling after hearing the name phonetically. The same people would sometimes merely adopt that spelling of the name on their documents from then on. So in researching the name be prepared for spelling changes in the same family line. I should add that sometimes the same clerks would estimate ages and this can cause inconsistencies between birth records and passenger list ages.
ANCIENT HISTORY OF THE NAME FITZSIMONS:
The Irish Family Fitzsimons emerged as a distinguished family in County Cavan (Ireland).
In 1172 the Fitzsimons accompanied the De Courys in their conquest of Northern Ireland and settled in Cavan. Others joined the Prendergasts and made their home in Mayo in the early 1200's (see the name "Eddery" below.
In 1323 the main Fitzsimons family came from Herefordshire England to Leinster, Ireland. They occupied territory in the Pale just outside of Dublin. It is now mainly found in Counties Cavan and Down IF291; MIF257; Map Westmeath. See under "Kimmons".
Eddery: "Mac an Ridire" (ridire = knight) meaning "sons of the knight". An irish patronymic assumed by the FitzSimons family of County Mayo during the thirteenth-century.
Kimmins: An occasional variant of Cummins in Counties Monaghan and Armagh where it is usually "MacCoimin" (not Mac Shiomoin) which is FitzSimons.
Another distinct group of FitzSimons occurs in Leinster from the fourteenth century onwards. In the sixteenth century the name principally occurs in several Leinster counties and in County Down, and today the surname is principally found in Ulster and Leinster.
Walter Fitzsimmons became the Archbishop of Dublin in 1485. The Fitzsimons name was also found in the Four Masters in 1505 as represented by Edmund Fitzsimon. Sir Richard Fitzsimon was one of the founders of the Knights of the Garter. Notable amongst the family up to this time was Sir Richard Fitzsimon.
In 1845 the Great Potato Famine culminated several years of famine causing widespread poverty and starvation and the great exodus from Ireland began. Within fifty years the population was reduced to less than half.
Many joined the armada of sailing ships which sailed from Belfast, Dublin, Cork, Holyhead, Liverpool and Glasgow all bound for the New World. Some called these small ships the "White Sails", others, more realistically, called them the"Coffin Ships" voyaging across the Atlantic when 25% of the passengers died at sea.
In North America one of the first migrants which could be considered a kinsman of the sept Fitzsimons, of that same family was Thomas Fitzsimmons who settled in Maryland in 1776; P.Fitzsimmons settled in Charleston SC in 1821, Edward Fitzsimmons settled in New York in 1822, Michael Fitzsimmons settled in Boston in 1850; Henry, James, Joseph, Patrick, Thomas, Walter and William Fitzsimmons all settled in Pennsylvania between 1854 and 1856.
Thomas Fitzsimon settled in Pennsylvania in 1845; James Fitzsimons settled in New York in 1853.
Many moved westward with the wagon trains and settled the mid west, some trekking over the Rockies to the west coast. Some remained loyal to the crown during the American War of Independence and moved North to Canada, becoming known as the United Empire Loyalists. Others formed the Irish Brigades in the great struggle for independence.
In current history many prominent people represent the name Fitzsimons, William Fitzsimmons an MP from Northern Ireland; and Frank Fitzsimmons, an American Labour Union President.
Further information can be found here