We went summer 2022 and really enjoyed it. But even in July the weather was a bit damp and cloudy a lot of the time (apparently that was worse than average, but I think typically dry sunny weather there only lasts two months). We did a week on San Miguel with our 18 and 20 year old, then a second week (we had to fly) on Terceira just us. We probably preferred Terceira - we thought the main town was nicer - but there are probably more activities on San Miguel.
Accommodation and food are really good value. If I could find inexpensive, convenient flights (for Terceira in particular) I'd definitely had back for a inexpensive, active but relaxing, beautiful, not too hot, summer week again.
We did lots of things with the (grown-up) kids: Canyoning, whale/dolphin watching, obviously the hot springs at Furnas, walking near Sete Cidades, beaches - including this amazing place where the hot water goes into the ocean so you get a kind of outdoor hot tub, a volcanic larvae tube underground, furnas volcanoe museum, kayaking, some great walks: we particularly enjoyed an unusual walk at Rota da Agua with some quite scary tunnels!
I think it is a great place with a 13 year old boy if he is likes outdoors stuff.