O'Brien search

My great-great-grandfather, Stephen O'Brien, arrived in the United States around 1870 and the only records I have for him are the ones generated in the U.S.  Unfortunately, these records only state that he was from Ireland.  The U.S. 1900 census documents that he was born in December.  After searching Baptism records in Ireland, the only Stephen O'Brien who matches a December birthday from 1840 through 1842 (which I used as estimate years) was born in Galway.  Having said this, I am only guessing the year and the county.may be correct.  Also, I cannot determine when he arrived in the U.S. because he had a very common name.  The 1900 census also states that he was naturalized.  I have found an 1869 naturalization document on Ancestry of a Stephen O'Brien who was naturalized in Connecticut in 1869, but again, I cannot be sure this is him.  I would appreciate all help and suggestions.  Thank you in advance for you time.

Cindy O'Brien Lee