Halliday is Mary's married name. I haven't figured out her maiden name yet. I have a guess about her husband - but only a guess at this point - I am hoping that the census can confirm that her husband was John Halliday, born 1831 Scotland. The 1861 census shows a John Halliday at sea, age 30 and married. With two of their children born at sea, it seems like a good possibility.
As for their children, here is what I have:
Emily, born 1849 at sea
Alice M., 1855 Poplar, London
Mary J., 1858 Poplar
Florence, 1860 Poplar
Arthur A., 1861 at sea
Ada E., 1864 at St. Botolph's, London
The 1881 census shows Mary born in 1828 in Rochester, widow, living with her daughter Alice (Caines) in West Ham, Essex.