I think, in general, the barbarian-ish Irish with their poxy little kingdoms are too lowly to be married into the direct ruling line. Especially considering the Norman focus was always on the continent and it's natural for them to seek alliances there.
Now, breaking the continental focus might make the Kings of England focus more on Ireland, but that's extremely hard to do- IOTL they focused on French territorial claims for, like, four centuries (as attested by Henry VIII's French campaigns) despite numerous catastrophes, especially in the reigns of John and Henry VI.
It's obviously post-house of Normandy, but in the 1150s Henry II supposedly wanted to invade Ireland and make his brother William king.
So, if you get a longer-lived House of Normandy (easiest POD is avoiding the White Ship disaster and/or having William Adelin survive it) you could see a similar scheme where a younger brother/son gets set up with a lordship in Ireland (or they can do it independently of the king). Then, they could marry an Irish heiress to claim lands on her behalf/pseudo-legitimise their rule- which happened OTL when Richard 'Strongbow' de Clare married a daughter of the King of Leinster.
Then, if you want Irish-blooded Kings of England, you have fortuitous deaths that see the kingdom of England fall into the hands of our hypothetical younger brother and his Irish wife.