Friends from near Limerick who are visiting right now recommend stopping in Newmarket on Fergus or especially Ennis for breakfast.
Our friends also recommend the restaurant in the airport if you are hungry coming off the plane. They just ate breakfast there after checking in for their flight here.
Are you and your family the type that want to get to your accommodations as soon as possible or want to really stretch your legs (and not sit) after the flight? (My husband is the former, I am the latter, but what we do is determined by the needs of our pre-schooler.)
If you're a "Stretcher," I would recommend walking around Ennis. Ennis was surprisingly charming, with a lovely friary, a nice riverwalk, a pedestrian area plus lots of restaurants. (You'll arrive early enough to get parking.)
Or if you have a really really good map or directions from others on TA, I would recommend exploring O'Dea's on the way there. O'Deas was fascinating and we did not allocate enough time since we had to meet up with our friends. O'Dea's would be first on my list for our next visit.
On the way to Doolin, you might want to stop and raise a pint to Father Ted in Ennistymon or Kilfenora.