The question is:
How would one convert
He said, "My name is Harry"
to reported speech?
Rules do say that the converted sentence reads
He said that his name was Harry.
The factor causing discomfort is the 'was' part of it. So it means that his name 'was' Harry and nothing is said about what his name 'is'! So there is a radical change in meaning!! In that light, the converted sentence should read
He said that his name is Harry.
Isn't the latter the correct conversion?