In my surname studies over the last few months, I am coming to the conclusion that the name Finney has to be considered to be of English origin.
I know of at least two Finney lines that came to America from England in the 1600's and first few years of the 1700's (John to Massachusetts and Captain Samuel to Philadelphia).
Some Finney's came to North America from Ireland (Robert, to New London, PA), also around 1700, but my suspicion is that those family lines were transplanted generations earlier from England.
There were a number English, including some Finney's, from the midlands of England (Staffordshire, Cheshire, Lancashire) who had land holdings across the Irish Sea.
But if you go back further, they may very well be a "French Connection"... if you refer to the UK Finney webpage (http://www.finney.ic24.net/index.htm
where it is reported:
"We came across an article about the origin of the Finney name, in that it is derived from an
old place name in Staffordshire, spelt Fynney, which originated from John Baron Fenis,
kinsman to William The Conqueror who was the last chap to successfully invade England
when he defeated King Harold at the Battle of Hastings in 1066."
So it sounds likely that Fenis was a Norman. William the Conqueror granted some fief's to those who aided him at Hastings, so probably Fenis' land was in the area
which eventually became known as Fynney.