How about via expansion at a relatively early juncture, and a union of the crowns between the Kingdoms of Ireland and Wales? Just across the Irish Sea, easily defensible due to mountain ranges and natural barriers, Wales would be the first place on the list for a powerful Kingdom of Ireland to either conquer or merge with. And as with the Sudreys, the Irish established large colonies here IOTL, primarily on Anglesey and the Lynn peninsula, before the Goddodin warlords pushed south, drove them out and established the Kingdom of Gwynedd, which would control the entirety of Wales and cement Wales' identity before finally falling to Edward I of England. Have the Irish either hold onto Wales or consolidate its influence there later on (King John I- of England IOTL, but whose father originally intended to make him the first King of Ireland before that- had made a treaty with Llewelyn the Great, and Llewelyn married John I's daughter, Joan, Lady of Snowden; in an ATL, perhaps there could be scope for a Union of the Crowns, paralleling that between England and Scotland IOTL, but preceding it by centuries?), and you've got another giant leap towards Ireland being a major naval and military power, as well as a massive step towards stymieing England's total dominance over the British Isles early on.
So let's say that, ITTL, in the late 1170s or early 1180s, Henry II manages to gain the approval of either Pope Alexander III or his successor (perhaps if someone more amiable than Lucius III gains the Papacy instead) to have John crowned King of Ireland? From that POD, John Angevin's first visit to Ireland in 1185 goes far better than it did IOTL, and John I is successfully coronated as the first King of Ireland, with him and his successors going on to develop and expand the Irish economy, increasing the efficacy of Irish taxation and established numerous new Irish market towns. In contrast, in this TL, the English economy largely stagnates relative to that of Ireland- John never becomes King of England, butterflying Liverpool (a town which he personally founded by royal charter himself in the early 13th century) out of existence, with far-reaching consequences for English colonial aspirations.
John's elder brother Geoffrey escapes his death in the jousting tournament in 1186, becoming King of England instead; Geoffrey's personal friendship with Philip II of France sees Geoffrey retain his place as the ruler of Normandy, Brittany and Anjou, as a vassal of Philip II; the long-proposed marriage of his daughter, Eleanor, to Philip's son and eventual successor, Louis VIII, goes ahead without King Richard alive to interfere, and the Angevin kingdoms on the mainland soon merge with the Kingdom of France. However, he becomes a far more neglectful and tyrannical ruler in England than John I ever was IOTL, rousing far more resentment about his even worse misgovernment, fiscal policies and treatment of many of England's most powerful nobles. As such, with the Magna Carta never signed ITTL, the First Baron's War becomes a full-blown civil war, one which ousts the Angevin dynasty permanently. Largely indifferent about the loss of England, with his primary focus on France, Geoffrey withdraws from England; and the Kingdom of England fragments, with the 25 (probably more ITTL) feudal baronies fighting against one another for dominance, and England becoming a patchwork quilt of principalities with no true king, to an even greater extent than Germany and the contemporary Holy Roman Empire.
Because of this, the primary base of power, wealth and trade in the British Isles shifts permanently from the chaotic, divided and war-ravaged lands of England to the stable, unified, well-governed and far more peaceful Kingdom of Ireland- with Llywelyn the Great, one of John I's greatest allies and son-in-law IOTL, along with the unified Kingdoms of Wales under his dominion, eventually becoming vassals of the Irish crown instead of the English crown ITTL, and with a dynastic union between the Royal Houses of Angevin and Aberffraw. And when British colonialism does kick off ITTL, instead of setting sail from Plymouth, Portsmouth, Bristol, Liverpool and London, the ships set sail predominantly from Dublin, Cork, Limerick, Derry, Waterford, Drogheda, and perhaps even Bangor, Anglesey, Pembroke and Chester as well. And TTL's British Empire effectively becomes the Irish Empire instead.
IOTL, Ireland came as near to matching England in population as it ever did back in the period 1400-1450; at this time, Ireland was actually more densely populated than England, with England's estimated population standing at 1.9-2M, and Ireland's estimated population standing at 1.25-1.5M. And if you added the populations of Wales at the time (i.r.o 300K) and the annexed Scottish territories which had formerly comprised the Kingdom of the Isles (i.r.o. 50-80K) to TTL's Kingdom of Ireland, making the Irish Sea a truly Irish Sea ITTL, then you'd have had the two of them basically on a numerical parity with each other back then, immediately after the end of the Hundred Year's War and prior to the Wars of the Roses. It certainly isn't ASB that a weakened England's population might remain on a par with, or lower than, that of a strengthened and expanded Ireland in an ATL, leaving the British Isles dominated by Ireland instead in an ATL.
It's also worth mentioning that we now know that the Irish got to both the Faroe Islands and Iceland first IOTL. But because the effort on Iceland was only mounted by a small and limited monastic settlement of hermits, they never truly colonized the island; the Papar instead merely resided there for a few years before abandoning it. And on the Faroe Islands, virtually all of the Norse colonial settlers which eventually came to dominate the islands, along with the effective leaders on the islands, came from the Norse settlements around the Irish Sea. However, an earlier attempt during the age when Irish pirates were the dominant raiders and settlers in the North Atlantic, immediately following the Roman departure from Britain in the early 5th century, or a later attempt in an ATL in which the Irish increase their dominance of the seas, and thus become true rivals of the Vikings, could easily see Iceland being permanently settled by the Irish instead of by the Norse as IOTL. And thus, you'd have an Irish naval empire, effectively keeping the Vikings contained within the North Sea and the Baltic.
And thus, from there on, all of OTL's Viking colonies or potential colonies further west, north and south are there for the taking for TTL's Irish. Such as Greenland, Newfoundland, Madeira and the Azores, and possibly even the Canary Islands. And from there, they could advance onward to North America, The Caribbean, North and West Africa, and potentially even South America as well, depending upon where their focus lies, which trade commodities and other opportunities they seek, and the levels of competition and adversity they face. If you want an Ireland more powerful than England (or any other nations/countries in the British Isles), then its greatest advantage would lie in its capacity to project power towards the west, across the Atlantic.