Just a rough idea:
Irish monks were notorious for trying to find lonely out-of-the-way places to build places for them to meditate and live the life of a hermit. There are some theories that the Irish made it as far north as Iceland before the Vikings and their monasteries dotted the continent, especially Gaul and Germany.
Now, lets say that some Irish monk gets the idea of go and try to convert the Moors of Hispania, or else he gets invited to set up a monastery in the Basque lands. Either war, after getting some fellow holy men together, our intrepid monk (lets call him Saint Kevin) heads south from Ireland with a small number of currachs. They get blown horribly off course and, after sailing trying to find land, they come upon the Azores. A miracle!
The men make way to the islands and begin to explore and find the land to be fertile and uninhabited. The fishing is good, and our holy men are able to regain their strength and even make repairs to their boats. Even better yet, the place is utterly remote and good for spiritual contemplation. When the time has come for the men to return him, Saint Kevin says that he wishes to stay. One or two monks agree to stay with him (much to Kevin's disgust, at first) and the rest set sail, saying that they would return as soon as they could.
A year passes, and then two, but in the second year a small group of large currachs are seen. The boats make landfall and are greeted by Saint Kevin and his co-compatriots. Kevin is esctatic, and talking about the visions that he and the other monks have had over the past two years. Kevin has become convinced that the islands were on the gates of Heaven itself, and he has taken to calling the lands Tir na nOg. This excites the men, and many more agree to stay, while others return to Ireland to spread the word of this mystical place.
Over the next few years, more and more monks come to the Azores and a number of small monasteries are set up. Some of the later voyages bring a number of sheep which thrive on the islands, cattle (the lynch pin of the Gaelic economy) are more difficult but not impossible. As the monasteries become more developed, even some lay people begin to travel to the islands in pilgrimage, and some agree to stay. The population is very small, of course, but stories of the land become well known in Ireland (rumors spread to the continent, and some Anglo-Saxon and Frankish pilgrims come, but there isn't anything there to draw their settlement).
Finally comes the Viking age and the chaos which it unleashed on Ireland. The Azores begin to be seen as a place that is safe for Christians and more monks, and some settlers, arrive. Finally, following a nasty battle, a defeated Ri takes his man and their families and sets sail for the Azores, hooping to find safety for his people. They arrive and this comes to be seen by historians as the first permanent settlement in the Azores. More would follow; mainly a trickle, but enough that the islands are fully settled by Gaelic Christians within a century or so.
So: there you go; Gaelic Azores
Not sure how realistic it is, but I think its the best chance you are going to get. The Azores are out of the way and not on any of the main shipping lanes of the era (there is a reason they remained undiscovered for so long), so the best you are really going to get is an accidental discovery. Having it being done by holy men is the best chance to have some stick around, since it would likely be seen as a good place for hermits and monks. Pilgrimages weren't uncommon, especially among Irish Christians, so its likely that, once a monastery is established it would draw people (most likely the rich) and a few might stick around to settle. After that, its not impossible that some king or another might see it as a possible refuge from the Vikings.