There are two sources of baptism and marriage records in Northern Ireland:www.emeraldancestors.com
free to view pay for recordswww.rootsireland.ie
pay website although a not for profit
organization. Small fees to view, more for the records.
After 1864 a civil registration of bmd's should be available. PRONI
Public Records Of Northern Ireland, Belfast for information
Do you have the children's name for Henry and Eliza, in order, oldest first. These names would probably have their grandparents names which would help to locate where they are from. The marriage certificate would contain the
father's name of both participants but also the place/parish.
In 1864 Armagh underwent a tax called the Griffith Valuation and at that time there were 66 Haire's and 132 Martin's.
You would to match up parish's the reduce the amounts.
You could use a simple version of Griffith Valuation
Henry Hare x 4 in 3 Parish's Drumcree, Kilmore, Seagoe
William Hare x 1, 3 x Haire Parish Kilmore, Newry, Shankillwww.failteromhat.com
which just gives Parish's and townlands
hopefully getting down to just a few parish's that Eliza Martin & Henry Haire have in common.
RC or Protestant? also makes a difference.
In an 1881 Commercial Directory there is a Henry Haire on West St. Portadown - a grocer?
Portadown was a Town in the Parish of Seagoe, Poor Law Union or Civil Registration District of Lurgan
In the british telephone book for Northern Ireland
there are just 3 Haire and 1 Hare but 35 Martin's including one in Portadown. All come with full address and post code.
You can look for gravestone inscriptions, there are some on websites and often libraries have copies of journals researched by local schools and heritage societies.
There is a UK company - expensive - that has inscriptions from 1900. It includes Seagoe. You would need to contact me for the name and website of the company as rootsweb don't like pay websites on the message email@example.com
PS Armagh and Seagoe are on the website www.irelandxo.ie