I don't think this helps you, but my great-great grandmother was Bridget Mahon, who was born and died in Woodford C Galway. It would seem that Florence could either be a daughter or granddaughter of one of Bridget Mahon's male siblings. Her married name was Lynch. Below is what I know about her siblings so it may be that Florance is the daughter or granddaughter of either Cornelius (Con) Mahon or Thomas Mahon. Perhaps that's a start for you to learn more.
Name: Patrick Mahon
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Birth:
Death: Occupation:
Father: --------------------------------------------------
Spouse: Mary Houlihan
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Birth: 1812 Death: Feb 4, 1894
Children
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1 M:
Birth: Death:
Spouse: Marriage:
Cornelius (Con) Mahon
May 1, 1836 Woodford, Galway, Ireland
Dec 16, 1889 Derrylaghan, Woodford, County Galway, Ireland
Elizabeth (Betty) Mahon
Feb 1, 1864 --------------------------------------------------
Portumna
Woodford, Galway, Ireland
Druminamuckla, Woodford, County Galway, Ireland
Woodford, Galway, Ireland
Woodford, Galway, Ireland
Mahon
2 F:
Birth: Death:
Spouse: Marriage:
Bridget Mahon
Feb 17, 1843 Aug 15, 1915
Martin Lynch Feb 2, 1861
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3 M:
Birth:
Civil Death Record
Thomas Mahon
Apr 12, 1851
Name:
Date of Death:
Age:
Parish / District:
Address: County:
Status: Denomination:
Occupation: Sex:
Graveyard Relationship:
Name: Address:
Pat 04-Dec-1878
76 WOODFORD
Derrylahan Co. Galway
Married
Civil Parish / District
WOOD RANGER Male
Informant Present At Death
Mahon Betty Derrylahan
AGE RECORDED AS ABT 76 YRS
1802
Dec 4, 1878 Wood ranger
Derrylaghan, Woodford, County Galway, Ireland
Bolag, Woodford