Restoration of Irish Monarchy

Technically speaking, Ireland has a lot of edge cases.

If you go full legitimist, yeah there's a few candidates. If you want to go off someone who has a Gaelic King to their name (not just descended but an actual king of the name) literally everyone in Ireland has at least one. My one got dethroned by Brian Boru for example, so you could just grab some Paddy McGee off the Liffy Boardwalk but it'd be sketchy. The third option is the populist or 'Napoleonic' route of having the king proclaimed by the people with the approval of the military.

However you do it, the only thing you really truly need to legitimise a king is the blessing of the Pope, as the Father of Princes, his word is final on whether or not a man who otherwise has no right to the throne can be proclaimed king. Famously, it was the Pope's word that basically kicked off the Carolingian Dynasty when the Karlings, who historically ran the entire country while the Frankish king did nothing, wrote to the Pope saying 'Hey, I basically run the entire country anyway, shouldn't I be king?' and the Pope was like, 'Eh, sure.' And a palace coup was a go. So if we put a king in place, who has the support of the military (security and stability) and the people aren't tyrannised by him and he supports the Church, (Christian Kingship) the only reason I can see the Pope not blessing his reign would be for political concerns in a geopolitical sense, not in the particular instance. It reminds me of the reason why the Pope didn't bless the Irish constitution, there was a few of them, but one I found most convincing is that the Constitution didn't outright make the Catholic Church the state religion. The Holy Father's interest is, as it should be, for the protection and promotion of the Church, if he sees a Christian Kingship of Ireland as being good for it, legitimacy should not be a problem so long as we have a good, or at least traditional Pope.

This person is the successor of James II, and therefore claimant to the throne of Ireland: en.wikipedia.org/wiki/Franz,_Duke_of_Bavaria

Some nationalists advocated making his grandfather Prince Rupprecht, King of Ireland.

In Brehon law, only 'rigdamna' could be kings. This word literally means made out of the stuff of kings and refers to all sons, grandsons, brothers and sons of brothers of anyone who has executed the office of king. Any of these could legitimately be appointed successor by the derbfine. Rigdamna dissipates after the third generation, so it can be argued that there is no legitimate successor to any royal Irish title alive today pursuant to brehon law.

As a republican though, I would put down any attempts at the above with bayonets. Be warned! ;-)

This is correct, though the entire succession system can be determined as being Tanistry, as Tanist was another name for the designated or heir apparent. Its where we get the title An Tainisté from.

Also the logic here is also worth noting that there are successors of the royal clans of Ireland still living abroad in Europe (one is actually a priest on mission in Africa last I heard), collectively known as the Council of Irish Chieftains and even held a tentative office in the Republic before the Republic found an excuse to exile them again because we can't have any of that uncomfortable monarchism in our nice shiny republic now can we? ;-)

Basically after the flight of the Earls, many of our nobles switched to some form or other of Primogeniture in keeping with the customs of the nobility of Europe at the time and many were even welcomed in other Catholic countries, my own old overlords of the O'Neils have a branch in Italy, which is why you have some Italians with the family name O'Neil going back centuries. Most of the current ones still do charity and philanthropy for Irish causes and the like and most seem to genuinely want a right of return, but either are not allowed by the government or some other reason is holding them back.

If they were to return, the modern Irish Law is a hybrid of old Brehon Law and Common Law (which differentiates us from British and Northern Ireland Common Law, spent three years in University getting a useless degree to understand all this hullabaloo), and common Law does recognise the right of hereditary titles, even if those titles don't directly correlate to property ownership (indeed that might even be a case in their favour, if they agree to being titular, which they previously did during their tenure under the Republic), so if the nobles were to return, we'd have our O'Neils, our O'Connor Don's and suchforth with which to re-establish the old Gaelic hierarchy and from which a new High King can be selected, either under the Brehon Laws that were not abrogated but which do allow for a new claimant to 'deemed worthy' of kingship, or under the custom of Common Law. Indeed, there was even a rumour going around that DeVelara even offered the crown of Ireland to the Jacobite claimant to the throne (the aforementioned Duke of Bavaria) who for one reason or another declined the kingship (although I heard other versions of the rumour stating it was another Jacobite claimant who was offered it, but I cannot verify), its hilarious to think that the only reason we are not a monarchy right now is because the guy they asked to be king said no, but lets not waste too many times on what ifs, but rather what may be.

Like I said, Ireland has a lot of edge cases, many of which people would be shocked. For example, Ireland is also home to the last hereditary Knight in all of Europe, some guy who traces his lineage back to the Normans or something (or he might be a later arrival, I am not sure) who's descendant still lives on his estate and has been nothing greater nor lesser than a straight up Knight. (Which is even funnier when you understand that historically, even though Ireland adopted Feudalism after the Norman conquest, we never adopted serfdom, and even the serfs that were made/or brought here eventually became free clansmen and Gaelicised along with their Norman lords, so we had Knights and Galloglasses roaming the island at the same time, even if they were not the same warrior), A funny story is that during the revolution, several IRA soldiers came to his door demanding seizure of his property, but because of some convoluted legal loophole recognised by the provisional government, he was allowed to keep his property. Its an interesting read. We also have a number of barons still resident in the country both north and south.

So if we wanted to, and I mean if we really wanted to, we could not only invite back the Chieftains to form a council of kings from which to select a new High King (either from among their ranks or selection of some other worthy person) with either a Tanistry or hereditary titleship for the High Kingship or some other selection method, but we would also have a legitimate basis from which to start up a legitimate order/system of knighthoods, both civic and military witha  pedigree if we can make that aforementioned Knight a grandmaster of some chivalric order. I would also integrate the old barons and some of the protestant lords of the north (they'll become Catholic eventually, this might even help hurry it along) as well as re-establish the prominence of clans and clan loyalties in the popular consciousness, its already there subconsciously anyway and everyone knows it, might as well be proud of the fact and Ireland becomes the most eclectic, interesting and absurd nation on the planet.

It'd be hilarious as much as it would be glorious.

When was the last Irish monarch?
It's a bit of a moveable feast as the Irish aristocracy and the church had no real problem with the English claim to the whole island since the 12th century.

Medieval historians regard the high kingship as an honoury position that no family was powerful enough to hold onto for long.
James the second mentioned above was fondly remembered by the defeated Irish as Seamus an Chac.
Of course the last legal constitutional king of all Ireland was the caddish King Edward Viii

I think if you are looking for a one island system yes.