What system design questions can I ask a Big Tech senior engineering manager in a reverse interview?

Asking system design directly in a manager call isn't something I'd recommend doing: it'd likely create a weird dynamic since you're trying to assert dominance on the manager when the dynamic is in clear favor of the manager (since they're the one hiring). If you want the discussion to be technical, focus on asking the manager high-level questions about the technical challenges of the org and letting the discussion naturally flow to higher or lower level discussions. Questions like:

• What was the hardest technical problem your org has recently solved? What made it so challenging?
• What are the technical problems you're looking to solve within this half or this year? How does the org prioritize engineering projects on the roadmap?

would be good. You want to get the manager to talk about their org and what they own & see if the technical problems the org is facing interests you. If you want to make a good impression, focus on showing that you can hold and guide a conversation and not on trying to IQ test the other person.

I would take a step back and assess what you're trying to get out of the "reverse interview questions". Showcasing your knowledge through asking questions usually doesn't end well and gets awkward really quickly.

Instead, show some of your past work (if feasible) and ask for their feedback on it. Writing an article on a particular topic on system design and sharing it with them ahead of the meeting is a great entry point since the actual components of a system are probably not feasible to show.

An alternative you may want to pursue is presenting your POV (e.g. what you found interesting, how you would extend the idea, where you've seen it apply) on a topic you think they are likely to be well-versed in (i.e. lookup their team and job titles to narrow down the scope). It can come from Uber's blog or any other publication in the industry really.

Overall, the key is to go in and have a conversation about topics you are genuinely interested in and want to know more about while presenting your own views on it as well. The mindset is to forget about the "transactional" aspects of the relationship as in, I need to do X, so he will do Y for me.