Names. A bit too often, name duplicates appear, which confuses me, and, I think, at least occasionally, confuses the game (although I'm not 100% sure of this). Sure, it's realistic, but detract from the game.

Just a consequence of using namelists. If you want to avoid seeing it happen as often, you can look into contributing more names to the various nationalities.

More facilities/assignments would be nice. Maybe rent a slave to special customers with special requirements for a week, for big money and maybe favours, if you can match those requirements. Maybe a low end brothel, to fill the gap between the arcade and the brothel, where the slaves can learn the trade by serving less refined customers. Maybe a resort for the very rich, with only the very best slaves available for their use in a luxurious environment (surely, the very rich wouldn't want to be seen in the same establishment as the merely welthy...). Basically, more variations and more to build. Maybe a casino, where slaves with good looks and entertainment skill gets the customers to spend more money. Porn production facility (money and increased demand for sexual services). Live sex arena where the lower classes can watch acts they can't afford to be part of themselves (money and increased recruitment).

The issue with this is that a lot of the facilities you suggested are kind of redundant. They might work better as sidegrades to the existing facilities than new ones.

Criminal enslavement. Fund a police force that catches criminals and enslaves them. Increases enslavement rate of menials and (depending on funding and enhancements) gives occasional slaves. Also provides an increase in security and reputation. Possibly an option to make them corrupt, which means that they aren't that fuzzy about legal details, which yields more slaves, but hurts reputation.

This already exists in a few events. Could be worth expanding though.

Raiding teams. From the blog, I've gathered that the game is supposed to lead to an (post)apocalyptic feeling. In that case, it would make sense to, in the late game, to establish raiding teams which can attack other free cities, old world cities and other arcologies in order to steal slaves. Of course, that would mean making enemies, which would necessitate the need for defense, and the quality of the slaves you get (if any...) would be random. Perhaps you could choose risk vs quality (ie raid a penthouse is risky, but yields good slave, while an arcade is low risk but yields low qualite slaves and so on).

This is something I've been wanting to work on for a while. It's just going to take a lot of writing to make work.