At my revisit to Case, I asked a lot about the PI's current/past collaborations (to get an idea of who does related work at the same institution so that I can talk to those people later!).
Also, even if you're decided on the school, I think it's interesting to find out how the researcher came to be at that institution, how long they've been there, what they like about it, etc.
Depending on how detailed the conversation gets regarding specific projects, I have asked about funding sources-- NIH, private sources, NSF? (I have been talking to engineers, so NSF is a viable answer to this question.) On one hand, it sounds a little rude (kind of like asking someone how much money they make!) but on the other hand, it's good to know what the funding situation is in a particular lab (especially if the PI is younger/has not yet secured tenure).