It's surprising to me that I have mentioned one island off the coast of Ireland (Garinish) without mentioning the two that I see on a regular basis: Lambay Island and Ireland's Eye.
We went out to Ireland's Eye on a boat from Howth a couple of years ago. Unlike Garinish, it's completely wild and desolate but it still makes a nice day trip.
We were in Howth last weekend and I took this picture of the island from the old (East) pier.