When they ask for ways you influence people

I can recommend two good books at least:

1. How to win friends and influence people by Dale Carnegie
2. Influence - the Psychology of Persuasion by Robert Cialdini

Timeless, very valuable advice. 

Beyond that, I have learned a few things over the years:

• Find out what is set in stone. Even if people say things like "All bets are off" or "We have to draw this on a clean slate". That is hardly ever true. Some things are always set in advance. Certain boxes in an org-chart are pre-filled with certain names. Some boxes only exist because there is a certain someone who needs a certain type of box. Certain suppliers are already selected, certain systems or methods will be implemented. Find out these things and you will see your room to maneuver. 
• Incentives matter. People, especially in large organizations, will pursue things with great vigor if it contributes to their individual goals and is rewarding for them (usually financially). It does not matter if it makes sense for the organization or the project. So to change their behavior you have to find a way to either change their incentives or have the new behavior positively influence their goal attainment. 
• Egos and personal convictions matter. The higher up in a corporate hierarchy you go, the more likely it is that you will encounter Type-A personalities with strong egos. Those do matter. Pamper their egos. Use words, frameworks or visualizations they like to use, even if there's technically a better one. One anecdote: On a project not too long ago, my job was to prepare a workshop for the entire management board of a Fortune-500 company. I had originally proposed an elaborate war-gaming over the course of an entire day. In the end, I got a time slot of 90 minutes (which, tbh, is A LOT of time from an entire board). But there is no way in the world you can do a real war-gaming in 90 minutes. But for some reason, the word "war gaming" had stuck. So during the prep, I had endless discussions about the new format, and why we were not doing war-gaming. No matter how many facts I could present that war-gaming cannot be done in 90 minutes, IT DID NOT MATTER. In the end, I resolved the issue by calling the new format something like "Accelerated War Gaming" but did not change one bit of the content. It was smooth sailing from there.
• Play indirectly: Again in large organizations sometimes the direct way to a goal is not feasible because you can't get enough support behind it. But sometimes you can get the necessary support for an initiative by proposing something completely different, which is so unattractive (--> see: incentives matter) to the necessary power-brokers that they throw their support behind your originally intended idea.  

Good luck,

Elias

Which company are you interviewing with?