Spontaneous Generation of a Surname? Pickham

I'm trying to track down the origins of my Pickham family name.  The first reference I can find is a Richard Pickham and Anne Shallow who gave birth to William Pickham in Enniscorthy on August 5, 1796.  It's noted in the Catholic Parish Record that the birth was "spurious".  I understand that to mean "illegitimate" and that either the parents were unmarried, or the church didn't recognize the marriage.  

My William Pickham was b. 1815/1816 in Limerick and died March 17, 1897 in NY, USA.  He married Ellen Morrison in 1846 in Doon.  I know his father's name was William, but his death certificate did not note his mother's name.   The Pickham name in general is, after 1796, very specific to the Limerick and Tipperary locations.  I believe, based on my contact with someone who still lives there, that there's only 1 family left with that name and he's as stumped as me as to the origins.  We have a pretty good list going of all the documented Pickhams, but nothing that gives any more hints about the origins.  

It's a very unusual surname around the world.  Of the few people with that name that I've found in the US (including my line) and in Australia, it appears that they all trace back to Limerick and Tipperary.   If found elsewhere, it's usually a misspelling of Peckham, which, I believe is English.  My line here in the US actually changed the spelling to Peckham by the 1890s.   I don't think they could read/write, so it may have just been left to the people doing the documentation, but it stuck down through the subsequent generations.    

Is it odd that there are no Pickham births in Wexford after 1796?  "Wickham" seems to be pretty specific to Wexford, could it be that Richard changed his name?   Even with the free records available this week, I'm not digging up anything new.  

I know it's long-shot but any and all theories about potential origins are much appreciated!

All the Best, 

Annie H.