You can't really go wrong here. I've eaten in almost every place there is to eat on all three Aran Islands and never had a bad meal. From the traditional "pub grub" to the cheap backpacker-oriented places, it's all good!
The main pub on Inishere is a good place to eat. The bartender is always friendly and the food is relatively inexpensive. I recommend the fish and chips, and possibly the chowder if they have any that day. The pub is also a place where tourists and locals alike gather to argue, share travel information, and relax. It’s kind of a meeting place.
Favorite Dish: Go for some of the "gotta have 'em" fish and chips. Fishing is the primary industry on the islands, so you have to have fish at least once.
If you're trying to travel cheaply, the best advice in Kilronan is to go to the Spar. You know, the big grocery store. Buy whatever you need here and then cook it back at your hostel. This is also an excellent way to make friends with the people you're staying with. One girl made me a cheese sandwhich when I was starving and now I am her slave for life.
Favorite Dish: I'd go with stuff like jelly and bread. Things that you can make really quickly.