Samuel is shown as a Constabulary Pensioner. Judging by the fact his wife and children were born in Donegal, he was evidently stationed there for some time. You would be able to get a copy of his service record from the police museum in Belfast (for a fee).
http://www.psni.police.uk/index/about-us/police_museum/museum_genealogy…
Samuel appears to have married Mary Jane Sproule, in Dromore West (Co. Sligo) Apr ? Jun 1886. Vol 4, page 87. The marriage certificate should give you his father?s name and occupation.
The Griffiths revaluation records for around 1911 don?t list Samuel Stinson. However they do list a William Steenson who bought plot 13 in 1910, a farmhouse, outbuildings and 12 acres.
http://www.proni.gov.uk/index/search_the_archives/val12b.htm
It remained in his name till 1923 when it changed to James Jardine. So possibly William died around that time. (There?s a death registered in Dungannon on 11.8.1925, aged 80, which could be him). This looks to be William in 1901 & 1911. I?d suspect Samuel was staying with him in 1911. So perhaps they were brothers?
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Clananeese/Castletown/1735144/
http://www.census.nationalarchives.ie/pages/1911/Tyrone/Clonaneese/Castletown/864216/
Birth registration only started in Ireland in 1864. Prior to that you need to rely on church records. I?d suspect that Samuel & William both originated in the Castletown area and would be inclined to search the local Church of Ireland baptism records for them and any other siblings. (Parish of Carnteel). They are not on-line anywhere so far as I can tell but there is a copy in PRONI (the public record office) in Belfast. A personal visit in generally required to access them. You should be able to cross check that you have the right family from Samuel?s marriage certificate which should give you his father?s name. (Expect the spelling of the surname to vary a bit in the various records).
You could try contacting the current Rector/Vicar at Carnteel to see if there are any Stinsons in the congregation today. The only problem is that they probably won't know their own ancestry back as far as you probably need. You would probably have to do the research from the mid 1800s to build forward to link with their knowledge.