Man, I used to have a few ideas about that but I can't remember any of them at the moment. The only one that comes to mind right now is Doki Doki Literature Club which focuses on the "normal" side of the story (club, interacting with the cast etc). I found it to be the more interesting part of the game and would love more of that.
I think there's a mod expanding on this aspect but I never tried it so I can't say anything about on that.

I'll try to remember some other spin-offs I used to dream about. That said, are we talking purely about VN spin-offs or do (imaginary) VN side projects for other media also count?

Tweenage me wanted an Ace Attorney spinoff focusing around Franziska. Hey, at least we got two around Edgeworth…?

Just wanted to say this is a great question and I feel like if this community had a larger user base it would get lots of fun answers.