As most people have said, the great issue is the lack of internal cohesion among the Irish. There are exceptions. Brian Boru or Tairrdelbach Ua Conchobair were both rather powerful High-Kings, and some sources say the latter attempted to institute male primogeniture. However they lost power after a generation. I'd say one method of achieving internal cohesion in Ireland is by screwing the Irish in the short term for long term gain. Have someone conquer Ireland almost in its entirety, form a ruling elite that makes necessary changes to law and succession to gain internal coherence, and then have the foreign elite gradually be subsumed by the wider Irish culture while retaining the crucial changes. So 100-200 years of foreign occupation only for the resurgence of a more native Irish state.
One issue is that you have to prevent Henry II from coming to Ireland and declaring himself Lord of Ireland. He really didn't want Norman lords holding land in southern Wales to grow outside his jurisdiction, in particular he didn't want Richard de Clare to declare himself King of Leinster. Several things could help this. Prevent the Papal Bull of 1155 that supposedly gave the Normans leave to conquer Ireland in return for making sure the Irish church started to follow the Gregorian Reforms. The Norman mercenaries could not be as successful, not forcing Henry II to ensure his supremacy.
Here's a scenario. The Normans still invade after being recruited by Diarmait Mac Murchada, but they don't have the fabulous success of overrunning the Norse-Gael towns of the southeast coast. Somehow the Norse-Gaels win. However the Norse-Gaels of Dublin recruit the Norman mercenaries themselves after winning, promising them lands if they fight for them. Somehow the Norse-Gaels don't fall into the same trap as all the other areas where Norman mercenaries arrives only to overthrow their employers. Dublin and the Normans conquer Leinster, with Diarmait becoming a vassal of Dublin. With the Normans defeated and under Dublin's control, Henry II decides to ignore the activities in Ireland and the Church is satisfied with promises from Dublin to enforce the Gregorian reforms, since the English Pope Adrian IV who signed the bull is long dead. Over a century or so fabulous success allows Dublin to conquer all of Ireland. The Normans brought castle building and stricter succession laws, which were adopted by the Norse-Gaels of Dublin. Eventually Normans and Norse-Gaels Gaelicize, only keeping the new laws that brought stability. So you end up with a largely Gaelic Ireland state maybe around 1300-1400.
Or a simpler solution is to have one French king decide in their feuds with England to strike the English Lordship of Ireland. He marries a daughter or a female relative to one of the Irish kings still fighting the Normans. France sends enough money and men for that Irish king to unite independent Ireland behind him firmly, and then to overrun the English positions in Ireland. If the Irish don't elect that king's eldest son by the French princess, the guy goes to France. He returns to Ireland with men and French support, conquering Ireland. He then adopts French laws of succession, keeping his power through French support.