I see from the 1901 census that the Hamilton family was Presbyterian. For births before 1864, and marriages before April 1845, you need to rely on church records, where they exist. Here are the details of all the Presbyterian churches in Ardstraw. There are copies of their records in PRONI, in Belfast:
P Ardstraw Baptisms, 1837-90; marriages, 1837-1939. MIC1P/50
P. 1st Castlederg (or Garvetagh) Baptisms, 1823-1985; marriages, 1854-98 MIC1P/73
P. Clady Marriages, 1845-71. MIC1P/50
P. Douglas Baptisms, 1831-64 and 1868-1946; marriages,1832-1936. MIC1P/50, 59
P. Drumlegagh (or 2nd Ardstraw) Baptisms, 1865-1986; marriages, 1845-1930. MIC1P/303
P. 1st Newtownstewart Baptisms, 1890-1903; marriages, 1845-1907. MIC1P/59
P. 2nd Newtownstewart Baptisms, 1848 and 1861-1905; marriages, 1846-1910. MIC1P/59
P. 1st and 2nd Newtownstewart Baptisms, 1903-85; session minutes, 1903-68. MIC1P/59; T/2563
Tradition was to marry in the bride’s church, so Patrick’s marriage may not necessarily be in the same church as his children were later baptised. None of the churches has records for the period when Patrick was likely to have been born so you are probably not going to find his baptism.
There are no Hamiltons listed as farming in the tithe applotment records for Milltown in 1833. That may mean that they did not live there then, or that if they did, they weren’t farmers. Patrick appears to have acquired plot 20 sometime after 1833, but before November 1845 when John married and gave Milltown as his address.
A death in Milltown would be registered in Strabane. I searched the deaths in Strabane 1864 to 1901. There were just 2, both registered in 1866. One was aged 67 and the other 74. You might want to investigate to see if either is your Patrick. Griffiths clerks were sometimes a bit slow to pick up changes in tenant, and so either could be yours. You can view the original certificates on-line on the GRONI website, using the “search registrations” option:
You will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate.
If Patrick or his wife died before 1.1.1864 (when death registration started) there may be no record to find, unless they have a gravestone.
The Londonderry Sentinel of 3rd May 1834 has a notice concerning the estate of the late Rev Stewart Hamilton of Milltown, Tyrone. The notice asks anyone who has any unpaid bills relating to his estate to contact Mr Colhoun in Strabane. The Rev Hamilton was the Rector of the Church of Ireland in Strabane, so I am not certain it’s the same family as yours. He died 5.1.1833, having been Rector there for 28 years.
PRONI also has a document D623/A/169/18 dated 29.10.1813 from the Rev Stewart Hamilton to the Marquess of Abercorn (the local landowner)
“Asking to be continued as sole tenant of the fields which his brother, James, and he hold jointly from Abercorn, if Abercorn persists in wishing to get rid of James Hamilton”.
PRONI also has D1062/4/A/41 dated May 1790:
Articles of sale between John McNeelance and James Hamilton, both of Milltown, Strabane, for premises in Milltown, Strabane, Co. Tyrone.