I have checked two sources for births/baptisms on-line. The Counties they don't cover are Derry, Donegal, Carlow, Clare, Kerry, Cork S.,Waterford (see library) Monaghan,
Offaly (Kings Co.)so I would concentrate to those other areas.

good websites for most county resources:

www.censusfinder.com - United Kingdom - Ireland and County
www.igp-web.com + County

Do you know mother's maiden name?

In Griffiths's valuations of Ireland you will find a full census substitute. www.failteromhat.com and see what Counties come up.

I did look in the UK census's of 1841 & 1871 where persons born in Ireland were pulled out from the full census.
Then I looked into the 1861 Census Manchester. There the birthdate for Matthew is 1822 and still nothing as above.

www.familysearch.org is a good website and has the 1881 UK census, but as you say he died at 54 and birthdate was guestimated at 1818 based on that. This website has a new PILOT program which is found under 'search records'. This would include all Ireland, but I didn't find anything there either so you could give it a try. If Eliza's first child was James and not her dad's name I would try that as Matthew's father's name?

I can't help you any further. You can google the other counties, they will have heritage/genealogy centers just not on-line.

Annette Code
'genealogy for fun'