In stitches is a relatively modern expression in English. It comes from the idea that a person is laughing so much that they can’t breathe properly and gets a stitch (a sharp pain) in their side. The term stitch for a pain in the side dates back to Medieval times in English.
Daniel Cassidy, author of the incredibly bad book, How The Irish Invented Slang, was in the habit of inventing ‘Irish’ phrases to explain common English expressions, even when (as in this case) the word already had a well-attested and reasonable explanation. His explanation for ‘in stitches’ was that it is from Irish staid aiteas, which he claims means ‘a state of joy’. The first thing to note about this phrase is that (as with the overwhelming majority of Cassidy’s so-called Irish phrases) it is completely and totally fake. There is no such phrase in Irish. There is not a shred of evidence that any Irish speaker anywhere has ever used this phrase. The second thing is that it doesn’t make grammatical sense. Staid aiteas means ‘state joy’. For it to make sense, you have to put the aiteas in the genitive – staid aitis. But Cassidy wouldn’t have known this because he couldn’t be bothered to learn any of the language before setting himself up as some kind of expert and publishing his crazy theories as if they were genuine facts.