Time of Travel for Least Rain

You're likely to experience periods of daily rain at any time of year, but given your options I'd choose May/June - historically one of the dryer times of the year. We've generally found that Spring rains tend to blow thru quickly, yielding plenty of sunshine after the passing showers. The countryside will be glorious as well,
By September our experience has been that things start to get a little gloomy and overcast - especially in the West of Ireland if that's one of your destinations.

Climate data agrees with May/June. Also, crowds are less of a problem in M/J. We were in Ireland in May this year and had one day where we had a short drenching squall, and 19 days of warmth and sunshine. Also only needed one dinner reservation.

Same thing happened in May 2016 and June 2019, with 2016 setting high temperature records. Maybe we're just lucky. Your results may vary!

Even if you do get some blustery or wet weather, the infrastructure is designed for it. Trails are graveled, roads have camber, restaurants have awnings. I carry a flimsy nylon raincoat that sees some use as a wind breaker and is total protection if there's rain. Rain won't be more than a slight inconvenience and an occasional scenery obscurer.

Days will be longer in May/June, increasing the potential for non-rainy sightseeing time.

We were in Ireland from 5/22-5/29 of this year. It did not rain at all the whole time we were there; it wasn't even overcast. The temperature was very mild and the only time we even needed a sweatshirt was when we went to Inisheer in the Aran Islands.

We were in Ireland most of May this year, including the last week, mostly in the south (Counties Cork and Kerry). There was one late afternoon/early evening where it absolutely poured rain, but it was dry the rest of the time. Apparently it was a veritable heat wave in Ireland this May - not scorching hot, but far warmer than usual. By comparison, when we reached London the first week of June, it had been colder than usual.

Five years ago, we were in the northern part of Ireland (including Northern Ireland), and I don’t think it rained once. Maybe we’ve just missed prolonged rain on 2 trips. I’d say it’s a toss-up; go when other factors work best for you, and have a good raincoat in case there’s some rain. Umbrellas have been available where we’ve stayed, if needed.

We’ve also had really good luck in May with little rain and as someone else noted I love the longer days. In late May it will be light until 10:00.

https://www.irelands-blue-book.ie/IrishWeather.html

According to that source April and June are the driest