Common names are always difficult because there are so many false positives.
With such a common name as Thomas Jones, I would not search public death records or look for the death certificate until I had the precise place and date of death or burial.
To attempt to pin Mr. Jones down, I would search in two areas:
First I'd try searching the obituary notices in the Birmingham newspapers. The reason why I'd suggest looking for the obituary is that it would give the names of all the next of kin as well as a bit of a biography, so you should quite easily be able to identify if it is the Thomas Jones you are looking for.
I am not an expert on British research, so I can't offhand give you the best sources of newspaper/obituary information for Birmingham.
But I could suggest:
- findmypast.com that has British Newspapers from 1710-1953
- The British Newspaper Archive
- Other Various Obituary Records Searches
If you can find the obituary, you should have the information you need that will enable you to find his death certificate.
If you can't find the obituary, I would then try burial records and cemetery indexes, e.g.:
In addition to Birmingham and environs, you'll also need follow with a search of London as a result of your family 'legend'.
If you can find possible burial data or cemetery headstones about people that might be your Thomas Jones, the information could provide the clues you need to search further. You may then want to try the obituary search again and map each of them to the burial/cemetery people to narrow them down.
Supplemental information: Amy Johnson Crow just posted an excellent article: 3 Tips for Researching an Ancestor With a Common Name. In it, she says to identify your ancestor by many more attributes than just their name, and look for links to that information through that person's friends, associates and neighbors,