I agree with Posteria that you must get his marriage certificate to find out his father's name.
However, I think this family could be a good possibility. Look at how the name has changed over the years
1871 census.
RG10 / 5031 / 70 / 52
51, Commercial Road, South Shields
Barnard McIvor, head, m, 60, furnaceman at chemical works, Ireland
Mary do, wife, m, 58, do
James do, son, u, 21, blacksmith, Durham South Shields
Barnard do, son, u, 19, engineman at sawmill, do do
Henry do, son, u, 16, blacksmith apprentice, do do
1861 census
RG 9 / 3788 / 83 / 64
Laygate Road, Westoe, South Shields
Bernard McKeever, head, m, 50, Labourer, Ireland
Mary do, wife, do, 47, do
John do, son, 17, Durham South ShieldsAnn do, daur, 14, do do
James do, son, 11, do do
Bernard do, son, 9, scholar, do do
Henry do, son, 6, do do
1851 census
HO 107 / 2399 / 649 / 44
Slake Terrace, Westoe, South Shields
Barnard McHewer, head, m, 40, labourer, Ireland
Mary McHewer, wife, m, 38, do
Arthur do, son, u, 15, Tile maker, do
Hugh do, son, u, 9, scholar, Durham South Shields
John do, son, u, 7, do, do doAgnes do, daur, 4, do, do do
1841 census
HO 107 / 298 / Book 1 / Folio 15 / Page 24
Slake Gate Row, Westoe, South Shields
Barnard McKever, 25, Fac(tory ?) Lab, I
Mary do, 25, I
Arthur do, 5, I