Some time ago, I asked a question about a precursor for Robin named Batboy, see here. I've since found the following image that I think is the story I was thinking of and it turns out the character was called John Vance with the alias Batman Junior.
Can anyone provide anymore details? I believe it was a relatively minor story to the main story, so don't think it had a major emphasis on the front cover. I notice that Batman's bat symbol isn't placed within a yellow ellipse (as well as the general drawing style) which gives clues about when the story was written.