Edited for clarification:
We’ll be in Howth at the beginning of July, and we are thinking about taking the short 10–15 minute ferry ride to Ireland’s Eye, walk around the island, and see puffins!
My concern is that both my 15-year-old and I get seasick. Has anyone taken the ferry? Is the ride usually smooth, or can it be rough?
It it a bad idea?
My immediate thought was when did they stop landing ferry rides? The answer is that they haven't- they are exclusively provided by an almost identically named company- https://irelandseyeferry.ie/
I haven't done the trip, so can't answer the OP's question, but do know that landing cruises happen.
It sounds like a "don't miss" adventure to me! With that said, I would take something for the ride. I have done many similar rides in the ocean and gotten sick enough times (not always) that I now err on the side of caution.