How can I obtain a document that will have his father's name on it i.e., - Birth certificate or some other perhaps?
You have already sent a hint about Harvey Lavery (his son - my uncle) Thanks so much. That was new info!
If you mean George Lavery 9H8C-T78, born 26 March 1868, his birth registration is available for free on the Irish government website irishgenealogy.ie.
According to the registration, his parents were John Lavery and Ellen Daley.