As of Thunderbird 31.5.0 this is still relevant, and my current solution is "File -> Save As -> File" and then pick file type "HTML" which does "what you expect". But, you have to bounce out to another application like Notepad or Firefox (view source) to access the HTML source rather than being able to use the built in Firefox text editor, which is not always the most efficient path to your goal.
All the previous answers (view -> view source, other actions -> view source, and Ctrl-U, which are all the same) do not seem to work, because as Tyler Rick correctly notes they present raw message with mime formatting (as well as message headers). And, in the current version, I don't even find View Source available on drafts. Certainly the "view source" implemented is a powerful feature as well, I just think it was that wrong default choice and should be an advanced option.