Tracey,
Your best bet I think would be to search the Mormon records. You may find that he included the name of his father and mother when he joined the Church. They like to Baptise Ancestors into their Church which is why they have such extensive records, and so he probably did a family search, or if not some of his descendants may have. They may have a record of his joining but whether they allow you access I have no idea. I expect there is a Family History Centre in Auckland and much of their stuff is on microfiche.
A formidable task to find a Dai Jones in Wales. Best of luck,
Kris