I had the same thing - in later years the name changed to Bingham but in the early Irish records they are all Bigham. I'm trying to sort out the Bighams in County Down and we're probably related as a result, since there were so few. I found some info that is quite interesting though I've not independently verified, which is explains "Bigham".

Found this info out on the Internet:
Bigholm - Beith
I'd like to know anything about the area known as Bigholm in the vicinity of Beith. I've seen a reference to the Bigholm hills, regarding a public trail and nearby grazing animals. It is probable that the surname Biggam or Bigham originated either here, or from the village Bigholms in Dumfriesshire, according to The Surnames Of Scotland by Dr. George F. Black. Any history of the name or area would be precious to know.
While some Binghams and Bighams undoubtedly have English origins, it must also be fact that there are those with Scottish origins. Robert Bell argues in his -The Book of Scots-Irish Family Names- that Binghams in Ulster are probably of Scottish decent; the name was being used interchangeably with Bigham and Bigam even in late-nineteenth century Ireland. As I noted before, Bigham or Biggam is a Scottish name, being derived from lands of the name in Ayrshire or Dumfriesshire.

Bigham — can be construed as "established land" from the two words, "big," a verb of the Scots tongue meaning "to build" or "to excavate earth (or to make habitable)" + "holm," a noun of a northwestern English dialect meaning a type of terrain, usually low-flat land near a river, but also related to holm in Old Saxon, meaning hill. From local origin in Ayreshire (Bigholm) or Dumfriesshire (village of Bigholms). Biggam, Bigholm.