Lis,
On the GRO website you can now search for births, and it tells you the mothers declared maiden name. You can then use that name to search for siblings.
Dialling in name Lawson for the relevant years, and mothers maiden name Studholm, with phonetic variations.
Thomas Lawson, Sep qtr 1848 South Shields, Mothers maiden name – Studham.
Ann Lawson, Sep qtr 1850 Durham& Lanchester, Mothers maiden name – Studholm.
Robert Lawson, Sep qtr 1852, Gateshead, Mothers maiden name – Studholme.
William Lawson, Dec qtr 1854, Auckland, Mothers maiden name – Studholme.
Dorothy Ann Lawson, Sep qtr 1857, Auckland, Mothers maiden name – Studholm.
Margaret Lawson, Sep qtr 1865, Newcastle, Mothers maiden name – Studholm.
John Lawson, Dec qtr 1868, Newcastle, Mothers maiden name – Studholm.
Elizabeth Lawson, Sep qtr 1870 Gateshead, Mothers maiden name – Studham.
Dorothy Ann Lawson, Sep qtr 1872, Durham, Mothers maiden name - Studholme
Pretty much matches the children in 1871.
its really lucky that Anns maiden name was Studholm as thats fairly rare in Newcastle/Durham. She gives her place of birth consistently as Murton which is just outside of Appleby.
Alan.