Fingers crossed that you're offered the role and congratulations on what sounds like being really successful in the interviews!
You want to give as many concrete pieces of evidence of why you should be compensated higher as possible. It's not enough to say that you want more, you need evidence to justify why you should be given more. This could include:
- Your current compensation. Would you be taking a compensation cut? If so make sure that they're aware of this.
- Competing offers. Have you received any competing offers? If not, were you in salary negotiations with any other organisations that you could use?
- Benchmarks. Do you have any benchmarks from similar roles in other organisations that you could share as?
Remember, the more someone likes you and trusts you the more they are going to go to bat for you. I'm sure you would be but be courteous and friendly - it goes a long way. And make it clear that you can be won over - they won't want to help if they feel that you're not serious about joining.
There are loads of great resources about negotiating job offers - I particularly like this one from the HBR. https://hbr.org/2014/04/15-rules-for-negotiating-a-job-offer