Hi,
Ontario Deaths records - available on familysearch, Ancestry, microfilm etc,. - start in 1869, so your Harriet just missed being included in these records.
I checked the Ontario Cemetery Finding Aid (vbain posted the link) http://www.islandnet.com/ocfa/search.php
There are several people with the surname James in the Bowmanville Cemetery. There is a Thomas, but not a Harriet. This doesn't necessarily mean that she isn't there. Not everyone shows up in this index.
If you need a look up of the results of your search, try this
http://www.wightman.ca/~dkaufman/and scroll down to Durham County. There is someone there who will do lookups in the Bowmanville area.
The OGS has transcribed the Bowmanville Cemetery records. A few are free to view, but the others have to be purchased (or viewed in your library - if it has the books, or has access to the Archives of Ontario microfilms via interlibrary loan). It is a lot to browse through!
http://www.ogs.on.ca/durham/cemeteries/bowmanvl.htmhttp://www.archives.gov.on.ca/english/archival-records/interloan/cemetery-durham.aspxScroll down to Durham
You could also try the Bowmanville library. Now, the area is called Clarington!! They have an Ask A Librarian feature. They would likely have the cemetery transcripts. Since you know the date she died, perhaps there was an obituary?
http://www.clarington-library.on.ca/The Oshawa library too.
http://www.oshawalibrary.on.ca/Maybe someone on this list has access to the transcripts and can do a quick look??
Just some suggestions. I will keep looking though.
RK