Hello Geri,
The site
https://www.rootsireland.ie/ offers you an opportunity to do searches and an option to add parental details in order to reduce the possible matches . Once you have a possible you should visit the individual county page from the map on the home page. The returns I got from a free search :-
DANIEL 1869 Parish: CASTLEREA (RC) PARENTS: PATRICK AND MARY
DANIEL 1869 Parish: BALLINLOUGH (CP) PARENTS: PATRICK AND MARY
DANIEL 1874 Parish: CASTLEREA (RC) PARENTS: PATRICK AND MARY
DANIEL 1874 Parish: BALLINLOUGH (CP) PARENTS: PATRICK AND MARY
MARTIN 1865 Parish: CASTLEREA (RC) PARENTS: PATRICK AND MARY
MARTIN 1861 Parish: CASTLEREA (RC) PARENTS: MARTIN AND MARY
MARTIN 1862 Parish: CASTLEREA (RC) PARENTS: JOHN AND ELIZABETH
MARTIN 1879 Parish: AUGHRIM (RC) PARENTS: JAMES AND BRIGID
PATRICK 1878 Parish: BALLINLOUGH (CP) PARENTS: PATRICK AND MARY
PATRICK 1878 Parish: BALLINLOUGH RC (RC) PARENTS: PATRICK AND MARY
PATRICK 1873 Parish: BALLINAMEEN (CP) PARENTS: PATRICK AND MARY
PATRICK 1867 Parish: CASTLEREA (RC) PARENTS: PATRICK AND MARY
I would suggest you start with the least common Christian name, either Daniel or Martin and use a +/- 10 year period from 1874 and progressively narrow your search. You can further narrow the search by identifying a church baptism (RC) and what was also obligatory a civil record of a birth by identifying the registration district (CP) within what are loosely called Civil Parishes. The Civil records at LDS.org will identify Civil Registration Districts(CRD’s) which may differ to those at rootsireland only because rootsireland have broken the CRD’s into their respective sub districts to aid research.
From shipping lists there appears to be two Daniels of interest on the Ellis Island site. One aged 29 , a collier travelling with a wife Eliz aged 24 arriving May 25, 1895 and another aged 23 , a carpenter travelling with other O’Brien’s arriving Mar 30, 1896