In Griffith's Valuation (mid-1800's), there were four Noone tenants or landholders listed in Kilcolman parish (Claremorris), whom you can see listed at this link:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNam…
Even if your Patrick had left by then, one or more of those men might have been his father or an uncle (note that one was also named Patrick).
Interestingly, there was only one Meath family listed in Kilcolman:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNam…
In case those Campbells were related to you, note that there is a chance that they were also from Mayo, and possibly from near Claremorris. I have Campbell relatives from the neighboring parish of Killedan, and the Campbell's go pretty far back in Mayo. In Griffith's Valuation, there were 128 of them listed in Mayo. Four were in Killedan, but none in Kilcolman:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNam…
The Clan Campbell originated in Scotland, and Campbell’s are found extensively in Ulster among the descendants of Scottish settlers brought in during the Plantation of Ulster in the 17th Century. However, a branch of the Scottish Campbell’s had come to what is now Mayo earlier in the Middle Ages as “gallowglasses” (gallóglaigh in Irish); foreigners who were brought in to fight for one side or the other in local wars. While the Scottish and Ulster Campbell’s would be predominantly Protestant (except as noted below), the ones in Mayo are descended from people who came before the Reformation, and they have largely remained Catholic (as my Campbell relatives did).
The Scottish surname is based on the Scottish Gaelic caim beul, which means “crooked mouth” (cam béal in Irish). In Ireland, surnames of Gaelic origin (as opposed to ones of Norman or other origin) were almost always based on the name, the nickname, or a characteristic of a real or legendary ancestor, and that was often true in Scotland as well. They can sometimes look very pejorative to us now, but were probably not meant to be back when they were created.
Some Scottish Campbell gallowglasses also settled in the Middle Ages in what is now Donegal, and there was an unrelated native Irish sept (or tribe) nearby, in what is now Tyrone, whose name was also anglicized as Campbell, but is based on the Gaelic name Mac Cathmhaoil, meaning “descendant of the battle chief”. These latter two groups would also mostly have remained Catholic, though some might have converted after the Plantation of Ulster.
There was definitely communication between the two parishes [Kilcolman (Claremorris) and Killedan (Kiltimagh)], because one set of my great-grandparents consisted of a husband from Killedan and a wife from Kilcolman, and one of their granddaughters married a man named Campbell in Killedan. Some Campbell's have also appeared as baptismal sponsors in my family in Killedan, and I have a good number of Campbell DNA matches (in Ireland and Scotland). I don't have any known Noon relatives, but some of them also appeared as baptismal sponsors, though in yet another nearby parish (Killasser), so I may have Noon connections. I and several second and third cousins in/from that area have had our DNA tested, in case you have as well and want to compare results.
The surname Noon, by the way, is Ó Nuadháin in Irish. The origin of the name is uncertain, but one theory is that it is based on the name of an ancient sea goddess, Nuadha. The Ó Nuadháin sept is traditionally said to be descended from Niall of the Nine Hostages and, as Roger said, is found all over Connacht. The surname Meath/Math in Ireland is mostly of Scottish origin, but there was an Irish sept which used the surname Mac Máighe (based on an old form of the Biblical name Matthew), and that surname was anglicized in various ways, including Mac Meath, Mac Math, and Mac Mawe. I don't think the name originated in Mayo, which could explain why so few were shown there, but people from other areas often migrated to Mayo, especially in the 17th Century,
In case you want to look for other relatives in the Kilcolman/Claremorris area, the parish records which are available online can be accessed at this link, and you can use the interactive map there to move on to nearby parishes as well (such as Killedan):
https://registers.nli.ie/parishes/1056
The names and boundaries of Catholic parishes and those of the civil parishes (which are administrative districts) are not always the same, but in the case of Kilcolman they are the same, and you can find more info about the civil parish at this link, which includes a list of the townlands within the civil parish:
https://www.townlands.ie/mayo/kilcolman2/
Since there appear to have been only a few Noon families in Kilcolman in the mid-1800's and only one Meath family (based on the info above), if you manage to find one or more baptismal records for Noon's or Meath's in the parish register, even if they are from a later period, the entries may indicate the townlands where the families lived, and you could then check out those townlands in that list mentioned above, to see where your ancestors may have lived.
