Starting off with the 1881 census:
Griffiths Arthur Head Married M 35 1846 Chain Manufacturer Rowley Regis Staffordshire
Griffiths Fanny Wife Married F 34 1847 Rowley Regis Staffordshire
Griffiths Fanny M Daughter Single F 12 1869 Rowley Regis Staffordshire
RG number: RG11 Piece: 2862 Folio: 51 Page: 40
Registration District: Dudley Sub District: Rowley Regis Enumeration District:
Ecclesiastical Parish: Civil Parish: Rowley Regis Municipal Borough:
Address: 10, High Street, Rowley Regis County: Staffordshire
I found a marriage on FreeBMD for Arthur:
Marriages Jun 1868 (>99%)
Adams Phoebe Leach W. Bromwich 6b 816
GRIFFITHS Arthur West Bromwich 6b 816
Johnson Mary Sophai P West Bromwich 6b 816
Johnson Mary Sophia P W. Bromwich 6b 816
KRINKS George W. Bromwich 6b 816
Smith Bryan Hinton West Bromwich 6b 816
But there is one too many men there

Looking at the image shows that the page has been mistranscribed, it should be 846 (I've put in a correction)
Checking that on FreeBMD comes up with the entry:
Marriages Jun 1868 (>99%)
BECKLEY Mary W. Bromwich 6b 846
Lloyd Edward William W Bromwich 6b 846
Westwood Fanny West Bromwich 6b 846
So I think that the Fanny unknown who married Arthur Griffiths is Fanny Westwood who may be the one mentioned at:
Births Sep 1846 (>99%)
Westwood Fanny Kings Norton 18 412
Any comments anyone?