Background.
Philosophy of teaching. Do they have anything set you follow or do you go at your own pace in terms of timing?
Are they more classical? Do they do anything with jazz/improv?
Do they do any music theory teaching or ear training with the lessons?
Just get an idea of their personality. That's a big one.
Do they do anything different for teaching adults?
Do they have any end-of-the-term recitals? Or yearly ones? I think those can be good. Or do they have any studio classes where all of the teachers students get together to play for each other?
How long have they taught? How many students are they teaching? (Any of the other students going on in music?)
Personally I'd look for someone friendly, open to your ideas. Knowledgable. Some with many students and at least some experience. Some performing too, but not a situation where the teacher is just teaching to make money -- If they have a piano pedagogy degree or music education that would be a good sign for them wanting to be a teacher (although for piano, I'd be more impressed with piano pedagogy over music ed).
On the business side, maybe ask how flexible they are about scheduling. The standard is weekly lessons, but I know one person who takes lessons every other week. Cheaper and it's an adult so it works.
A masters student or piano professor sounds good. The masters student might be more optimistic and fresh. And cheaper. The piano prof could go either way. Some are rigid. Some are more friendly, although I'm thinking of one with a friendly exterior who might just be teaching to pick up some extra cash.