Fitz Henry is a Norman name, brought to Britain by the Norman invaders in 1066.

 

It is a patronymic name (describing a person by who their father was) meaning 'son of Henry', although the original Normans would have spelled it Fitz Henri.

 

From here on I'm going to use the modern spelling: Fitzhenry

 

Popular legend says that all Fitzhenrys are descended from Henry 1 of England by his mistress Nesta of Pembroke. However Henry was also reputed to have 20 illegitimate children by various women.
There are further true patronymic uses of the name spontaneously arising in later records so it is unlikely that there is one true 'common ancestor'.

Norman/Platagenet nobles took the Fitzhenry name to Ireland in the 12th century with the invasion of Ireland. The Fitzhenry name has historically been associated with County Wexford where the Fitzhenrys had several castles and desmenes.

In Ireland, the name became more "Anglicised" to Fitzharry or Fitzharris. Both Fitzhenry and Fitzharris were used by the same families.

In England, the name became even more Anglicised to Harrison (literally the Son of Harry) and the Norman version virtually died out in England and Wales.

With the creation of the British Empire, the Fitzhenry and Fitzharris names spread to all parts of the globe with a British influence. The bearers of these names were mainly Irish soldiers in the British army, or Irish prospectors and settlers. There were/are branches in India, Australia, New Zealand, Canada and South Africa, as well as a small number in South America. Most bearers of the Fitzhenry and Fitzharris names in England today can trace their routes back to an Irish ancestor.

There is a large Fitzhenry and Fitzharris presence in the United States of America.
Numerically, the largest family tree in the USA is descended from Enoch Fitzhenry, a legendary figure who was allegedly captured by pirates and taken from Ireland to the USA in the 1770s.

There are also a large number of smaller Fitzhenry and Fitzharris family lines in the USA derived from those Irish leaving their homeland for a better life especially around the time of and  after the great famine for the 1850s.