1860 would be a parish record, you will need a parish or try searching www.rootsireland.ie
without one. You have the County at least and Peter was quite an unusual name for the times as was Cornelius his father. If you search for a Peter born 1860 parents Cornelius and mother Margaret
you will get a record of all the siblings as well. This is a pay website although it is a not for profit
Foundation. Pretty much the only way to find parish records on line until Civil Registration started in 1864. see www.familysearch.org
I see no children born to Cornelius and Margaret after 1864.
The Griffith Valuation is used as a census substitute and took place in 1856 in Galway and at that time no Cornelius was an occupier of land. If Cornelius was not married as yet then he probably still lived with his parents, usually on marriage would take on a house and land. Peter was unlikely his father's name either unless he was already deceased as that name is not in Griffith Valuations as yet. If a second son he could be named after his wife's father!
An earlier census substitute the Tithe Applotments 1823-1837 show no Cornelius Gavin's in their records, but 6 Peter Gavin's: 2 Tipperary; 2 Roscommon; Mayo & Dublin
1824 Directory - www.failteromhat.com
Cornelius Gavin, Cooper, little Cray-Lane
(farmers don't appear till the 1881-1894 directories
There is one Cornelius Gavin but he appears in the 1851 Limerick Griffith Valuation
Parish Tullabracky, with town land of the same name in the Poor Law Union of Killmalock.www.askaboutireland.ie
There is a Peter Whelan, Parish Killora, Townland Ballymore, Poor Law Union Loughrea, Co.Galway
tenant of Richard Rathbourne
24A house, outbuilding and land 2 acres 2 roods and 6 perches
24B land 2.3.36
Patrick Kearns, Parish Killora, Townland Killora, other info same, tenant of John Lambert
#3 house, outbuildings and land 8.0.8
Very disappointing on www.familysearch.org
except Peter's marriage 24 Sept 1884 to Margaret Whelon (not Whelan) could be an error, he is 26 and says born 1858 not 1860 and his death on June 6, 1906 in Waltham, Mass.
This makes me think that the siblings of Peter to be younger not older as most women had children
every two years, and so a sibling in 1864 should occur if Peter was a first child! I hope that makes sense. What are the names of the children that Peter & Margaret had in the USA.www.igp-web.com
Another source of information might be gravestone inscriptions. I have a few for Galway but don't see any there that might tie in. Some to the later +1953, as the previous graveyard was full.
Libraries often carry gravestone inscriptions as many projects have taken place to preserve the
inscriptions both school children, local heritage or museum societies.