First of all, Irish naming pattern is: 1st son after father's father, 2nd son after mother's father, 3rd son after father then on to great-grandfathers, uncles, etc. For girls it is: 1st mother's mother, 2nd father's mother, 3rd mother...
Another thing which can throw off the naming pattern is naming next child after a relative who's just died. Still think Florence is unlikely for woman born early 1800s.