Why did she want him to change his name?
My sense of Ayn Rand is that, among other things, she was a control freak. I remember, many years ago, I came upon an article written by someone who was, for a time, a member of the "inner circle" around Ayn Rand. He was married, and his wife was a believing Christian. He said that Rand insisted that he divorce his wife because of her beliefs. He refused and that was the end of his relationship with Ayn Rand.
Sometimes I wish I had printed and saved all of the articles that I have read over the years, so that I would be able to find them when I need to recall something I had read a long time ago. But, I didn't.
It was likely because she wanted to share her name with him,
Nathaniel bRANDen.