I think you might have had the right idea right at the start - advertising jingles.

short and easy to remember, easy to sing or whistle.

As he has some years on his back trying to recall some old ones might divert any nerves about the actual singing of the words.

If you can get him to actually start producing notes, fine, or you might only be able to use the rhythm of the phrases.

It will at least make a start.