I am interested in your enquiry because of the Suffolk connection, and also because I have been researching my own seafaring ancestors.
The Suffolk Records Office in Ipswich has burial records for most parishes.
However before you try searching there, I suggest you establish some firm facts about Samuel Thomas (or Thomas Samuel) Smith. (I take it 'Brickwall' is not part of his name?).
Do you have any certificates at all that bear his name or other details?
You say his 'partner' was Elizabeth, perhaps they were not married and the children would have had her surname.
Do you know the names of any of the other children?
It's definitely worth looking in the census returns, I had a quick look in the 1911 census and there is a Derbyshire (Shardlow) family with the right names for father, mother and daughter Elizabeth. Also you can search the 1881 census for free via Ancestry.
Regarding Samuel's marriage, there are some possible records on FreeBMD - e.g. Sep 1894 Samuel Thos. Smith married Elizabeth Washer in St. Saviour, Southwark (London) district. That would make sense if he was a mariner.
Also worth checking British Chaplaincy records for marriage or christenings, some of my mariner ancestors' records are at the British Chaplaincy at Hamburg. You can search for these - as well as many British parish records (including some Suffolk) at www.familysearch.org