Paul,
Regarding the origins of the name Barber, MacLysaght (The Surnames of Ireland) says: “Barbour, Barber ie le Barbier. In Dublin since the thirteenth century; now mainly in Ulster.”
if your ancestors arrived in 1200s then that was long before the Huguenots arrival. The trigger for the their departure from France was the revocation of the Edict of Nantes in 1685 by Louis XIV. About 50,000 fled to England (seen as a safe haven as it was also Protestant) and about 10,000 settled in Ireland, mostly in counties with a significant Protestant presence, for the same reason. Wikipedia says: “About 10,000 Huguenots are thought to have come to Ireland, mostly settling around Dublin, Cork, Portarlington and in Co Antrim and Co Down.”
What denomination were your Barbers as that might help clarify their origins? Obviously the Huguenots fled religious persecution in France so are very unlikely to have been Catholic in Ireland. They did set up a Huguenot church in Dublin but most gradually switched to Presbyterianism (it’s Calvinistic philosophy being fairly close to their own beliefs). Some became Church of Ireland (ie Episcopalian).
Looking at the 1901 census of Monaghan, I see 33 folk named Barbour/Barber. 7 were Catholic, 13 were Presbyterian and the remainder Church of Ireland. So the majority seem to have been settlers here at sometime in the past. Those Presbyterians could be descendants of Huguenot refugees, or they may just have come from Scotland which is where most Irish Presbyterians have their roots.