Hi
A death registration in the General Registry Office domestic indexes in the C19th really required a body. If he died off shore because he drowned it would need his body to have come ashore and to be identified. If it came ashore further down the coast or later it may not have been identified in which case the death would be registered as unknown.
If he died aboard the coal barge then the death should have been registered where-ever the coal barge next put into port. Because of the expense of trying to keep a body without modern equipment, you would expect the body to be buried locally, unless the family had sufficient money to have it returned to be buried locally.
There is a possible burial in Whitby further up the coast from Scarborough (20 miles)
2nd January 1859
Thomas Willis aged 18
Is this the family on the 1841 census? If so the 1859 burial would be about the right age.
HO107 325/10 folio 17
Markhams Passage Moulsham Essex
Elizabeth Willis 45 Widow
(crossed through)Mary Willis 15
Joseph Willis 15
Mark Willis 13
Thomas Willis 10
all put as Lab, all born in county
1851 census HO107 1775 folio 429
Moulsham Essex
Elizabeth Willis 58 Head Widow Felstead Essex
Mary Willis 26 Daughter Chelmsford Essex
Cecelia Willis 6 Daughter Chelmsford Essex
plus 1 servant
If you think the Whitby burial is a possibility it might be worth contacting Whitby registration office to see if they hold the death certificate.
http://www.northyorks.gov.uk/index.aspx?articleid=2861&contactid=5659Regards
Valda