(2-part post):
I tried to research this question last night but, after an hour or two, I'd not found anything which I was sure would satisfy your requirements, so I decided not to post a reply.
However, nobody else seems to be answering, so I thought that I'd try to give you a few pointers:
I based my research on the official Irish tourism website:
http://www.discoverireland.com/gb/
(Use the pull-down menus on the right).
For cheap, family-friendly accommodation, I searched on 'All Ireland' for 'Caravan & Camping'. I was hoping to find some sites which offered static caravans for hire, together with 'family fun' facilities. Unfortunately, it seems that very few caravan sites in Ireland have statics for hire; those sites with 'family fun' facilities seem to be solely for touring caravans.
If you'd consider a quieter campsite, this one has mobile homes (sleeping 5 or 6, take your own bed linen) for hire. There are no prices given. Phone for details:
http://www.discoverireland.com/gb/ireland-plan -your-visit/accommodation/listings/?fid=FI_350 25
There are 3 mobile homes for hire on this site:
http://www.discoverireland.com/gb/ireland-plan -your-visit/accommodation/listings/?fid=FI_350 75
(Phone for details).