A large number of my old favorite video games let the player change characters' names. It didn't matter whether or not the characters had personalities. This level of player control seemed to be more common in the '80s and early '90s, when story had to be conveyed through dialogue and sprites rather than spectacular cutscenes, and before voiced dialogue was common.
How much customization should the player or reader have? My best answer right now is to shrug and say "I dunno." When reading
Sweet Fuse a few months ago, I changed the main character's name to one that I like, and I still admired her attitude and intelligence. But when I was reading
Tears 9, 10 before then, it didn't bother me that I couldn't change any names.
By the way, for those who are interested, there's a
large database of gamebooks available. In other words, it is certainly not limited to the books marketed under the
Choose Your Own Adventure label.