Kathleen:
Welcome to Ireland Reaching Out!
The 1855-1856 Griffiths Valuation head of household listing for Tynagh civil parish showed two Flanagan records. By the 1901 census, there were no Flanagans in Tynagh civil parish. I did find two Thomas Flanagan baptismal records in Tynagh parish in 1821. Possibly one of them is for your ancestor? See below.
Roger McDonnell
http://www.failteromhat.com/griffiths/galway/tynagh.htm
First name(s)ThomasLast nameFlanaganBirth year-Baptism year1821Baptism date10 Dec 1821ParishTynaghDioceseClonfertCountyGalwayCountryIrelandFather's first name(s)TomFather's last nameFlanaganMother's first name(s)MaryMother's last nameLarkinRepositoryNational Library of IrelandNational Library of Ireland linkhttp://registers.nli.ie//registers/vtls000632961#page/1/mode/1up
First name(s)ThomasLast nameFlanaganBirth year-Baptism year1821Baptism date14 Dec 1821ParishTynaghDioceseClonfertCountyGalwayCountryIrelandFather's first name(s)NedFather's last nameFlanaganMother's first name(s)BridgetMother's last nameKeoneRepositoryNational Library of IrelandNational Library of Ireland linkhttp://registers.nli.ie//registers/vtls000632961#page/1/mode/1up