Hi Philip
I hope if you haven't already that you get an opportunity to visit Achill; you will be happy that your ancestors lived in such a beautiful place.
I'm afraid I couldn't find a birth or marriage record for Neil or the boys on www.rootsireland.ie/ or findmypast.ie
I'm not sure which sources you are aware of but the Tithe Applotments (free at http://titheapplotmentbooks.nationalarchives.ie/) have 20 Gallaghers in Achill in the 1820's
The book Achill Island by Theresa McDonald mentions that the commonest surnames in the village of Achill prior to 1776 were Gallagher & Grealis.
I had a look on the cd Memorials of the Dead Galway & Mayo but couldn't find Neil or Bridget.
The early Irish Censuses (1821, 41 & 51) don't have any Gallaghers in Mayo!
Griffiths Valuation (free at http://www.askaboutireland.ie/) has 375 Gallaghers in Achill in 1855 including a Neill
Griffith's Valuation Record Information
TenantFamily Name 1GALLAGHERForename 1NEILLLandlordFamily Name 2MCCUEForename 2PATRICKLocationCountyMAYOBaronyBURRISHOOLEUnionNEWPORTParishACHILLTownlandDOONIVERPlace NameDOONIVERPlace TypeTOWNLANDPublication DetailsPosition on Page29Printing Date1855Act15&16Sheet Number43Map Reference5
I notice this is different to the one you have on ancestry however
I thought this might be Bridget
Name:BridgetGallagherDate of Death:09-Sep-1884
Age:67Parish / District:ACHILL DISTRICTAddress:Achill BayCounty:MAYOStatus:Widow (Female)Denomination:Roman CatholicOccupation:FARMERS WIDOWSex:Female
GraveyardInformantGraveyard:
Relationship:Daughter In LawParish:
Name:Margaret GallagherCounty:
Address:Achill Bay
Not much luck on findmypast.ie except for this:
NealLast nameGallagherYear1848Petition dateOctober 19, 1848/October 23, 1AddressIsland of Achill, Co MayoPlaceMayoCountryIrelandPetition titleAchill Island Co MayoArchive referenceCRF 1848 O 16/2/098Sheet Number4Record setThe William Smith O'Brien Petition 1848-49CategoryMilitary Service & ConflictSubcategoryCivil war & rebellionCollections fromIreland
Bridget would only have been 13 when she had John which seems very young.
I'm afraid my findings are insignificant Phillip!
Col Cafferky
Achill Liaison