Unfortunately, you have run into the same road block as everyone else who has attempted genealogical research in Ireland.
I can only offer that the name McGivern (various spellings) seem to be primarily in Co. Down in the Freeholder Records available at PRONI (Public Record Office of Northern Ireland)'s website.Michael would have been too young to be listed, as these records date from about 1820 to about 1830 (give or take a few years), but the townlands named include Loughriescouse (var. spellings), Cloghog(e), Lisslea (where a Michael McGiveran is noted), Ballynanny, Cargins, Drumgreenagh, Cregans, Mullaglass, Greencastle, Ballyboley, Derrymacash, and Drumsisk.
What were his children's names. The naming convention will generally lend some clues as to the names of the parents.