Civil registration started here in 1864 - so you might find details of Michael's birth. Familysearch have an online index to the registers at :
http://pilot.familysearch.org/recordsearch/start.html#p=2;t=searchable;c=1408347I dont see a good match in at the moment in 1864, the closest being this record in 1865
Birth
Name: Michael Broderick
Registration district: Loughrea
year: 1865
Volume: 14 / Page: 386
(There may be other matches with spelling variations..)
you can order research certs from the GRO (
www.groireland.ie) using the index details.
The index lists registration districts - these are the districts in, of partially in Co. Galway :
Ballinasloe, Ballinrobe, Clifden, Galway, Glennamaddy, Gort, Loughrea, Mountbellew, Oughterard, Portumna, Roscommon, Scarriff, Tuam
A cert will give you a townland or town which should provide a stating point for further parish searches.
Shane