I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are sparse B4 1840, rare B4 1800.
Irish usually emigrated between 16 & 24 yrs old. Assuming by the turn of the century to be between 1891 and 1909. This infers a DOB between 1867 and 1893. This wide net catches 9 B rcds for Thomas Conboy B Roscommon. Some of these may be duplicates.
B/M/D registration was civil after 1863. Civil rcds are kept by the General Register Office. Inquire at http://www.groireland.iehttp://www.groireland.ie. You can also use the IFHF database, but its Mayo rcds aren't quite complete.
Google the Foundation's name to lead you to the database. Register, opt to be notified as rcds are added and browse. Pls post you search results to my genforum msg.