I have come close to finding a connection to Babingtons in Donegal, Ireland, through middle names.
My uncle was an Andrew Babington MacAuley, born in 1915 at Bomany, Conwal Parish.
His grandmother (my great-grandmother) was a Jane Hamilton born 1849 at Drumbeg, Taughboyne Parish.
Jane's father was Robert Hamilton of Glenfad who married Margret (Margaret?) Devenny of Drumbeg, Taughboyne in 1845
Robert Hamilton's father was James Babington Hamilton. noted as James 'Robington' Hamilton on the 1845 marriage register,almost certainly incorrectly because they named one of their sons born 1852 'James Babington Hamilton'
Any information on the 'Babingtons' would be of interest.