Most anglicised forms of Irish surnames, even some apparently of the "hard" crossword variety, can be riddled, and while we are not informed of the relationship of Biggar/Bigger with Biggerstaff, we can see a connection.
Puzzling indeed is the information in Mac Lysaght's Irish Surnames that some Co Down Biggars and Biggerstaffs are actually of the sept Mac Givern. And even more puzzling is that Montgomery, another Scottish name of Norman origin, has since around 1900 been used interchangeably with MacGivern.
The place Bickersteth ("beekee pers" staithe/landing-place) became the surname of those who dwelt there. This changed to Bickerstaff(e) and then, betimes, to Biggerstaff. The Phone Book of Northern Ireland lists 62 Bicker staffs, while south of the Border there are three Bickerstaffes, all in Cork. No Bickersteth, though in 1876 there was one in Land Owners of One Acre and Upwards.
That was Mrs Bickersteth with 74 acres at Lislea, Kenagh, Co Longford. There were then seven Biggar/Bigger holdings - in counties Louth, Antrim, Armagh, Donegal, Derry and Tyrone, ranging from the 1,274 acres at Falmore Hall, Dundalk, to the 36 acres of a second resident there.
The second largest holding was the 773 acres of Joseph Bigger, Trainfield, Belfast, who also possessed 43 acres in Co Donegal.
Odhar means "dun, dun-coloured, pale, wan, brown, khaki-coloured, weather-beaten, sallow", and was used as a personal name, which may mean dark, sallow, grey-brown, but it is possible that here we may have the ancient name of the otter used as a personal name.
This is a relatively rare early name. Among its bearers is Odar Mac Muiredaig, abbot of Lusk, who died in 1055 (Irish Personal Names, O Corrain & Maguire). From this we get the surname Mag Uidhir, "son of Odhar', anglicised MacGuire/Maguire, and O Duibhir (dubh, black, and odhar), anglicised (O) Dwyer.
A diminutive of this is Odhran, the name of 17 saints, according to an early text, and from this we have the surname O h-Odhrain (Horan). The Co Wexford surname Bolger is from O Bolguidhir (bolg, belly, and odhar. Yellow belly!),
Another diminutive is uidhrin, from whence O hUidhrin, (descendant of Uidhrin), anglicised Heeran/Heerin, the name of an Offaly sept, and Mag Uidhrin (son of Uidhrin), "son of little Odhar", anglicised Mac Givern. Annala Rioghachta Eireann/Annals of the Four Masters notes the death in 1120 of Eachmarcach Mac Uidh rin, chief of Cinel Fearadhaigh.
The name of this ancient territory was anglicised Kinelfarry, and was in, or near, the Co Tyrone barony of Clogher. Owners shows a 17-acre M'Giverin holding at Dunmanway, Co Cork; a 50-acre M'Givern holding at Dungonnell, Crumlin, Co Antrim, and holdings of four and 13 acres at Ballingtaggart, and Cappagh, Co Down.
Of the 234 listed in the telephone directories of the island, 155 are north of the Border, largely in Co Down, particularly around Warrenpoint. Those south of the Border are concentrated in counties Louth and Meath.
McGivern was among the many names of families who had been tenants of church lands inherited from the medieval church by the Church of Ireland bishops of Dro more, listed in the 1609 Schedule of Iveagh. The McGiverns appear to have held Ardbrin, Lisnasliggan and Tullintanvally in the Co Down parish of Annacloone.
These families - McGivern, McBredan, McBrin, McAteer, McCornick, McDonegan, Mc Ginn, McGowan, McKey, O'Kerny, O'Roney and O'Shiel - appear among the medieval clergy, and in place-names in this diocese of Dromore.
To the north of the parish of Annacloone, wherein once the McGiverns, is the parish of Garvaghy. Herein is a townland named Knockgorm.
"The present form of Knockgorm might well lead one to assume, with O'Donovan, that it represented Cnoc Gorm `blue hill'," says Volume IV of Place-Names of Northern Ireland, which covers north-west Down/Iveagh.
The conclusion reached in this work is that, strangely, this derives from the Irish Cnoc Mhig Uidhrin, "MacGivern's hill".