What are the best arguments to start using CiviCRM
That depends on what they're comparing CiviCRM to.  If they're comparing it to a current system of legacy software, Excel sheets, Google Forms, etc., that's different from someone who's comparing it to Salesforce or one of the other more serious competitors to CiviCRM.  It also depends on what they want to use it for - pitching CiviCRM is very different for folks doing grassroots fundraising vs. case management.

To respond to some of these scenarios:
There's a YouTube video of my lightning talk from CiviCon 2015 where I talk about CiviCRM vs. the competition here: https://youtu.be/TfIzV1sj9qY?t=14m34s.  It's 5 minutes long - I meant to write it in a blog format, but haven't yet.  In short: It's stories of how the other major vendors have screwed users over by jacking up license fees, created higher-end products at much higher prices than whatever they sell folks originally, or refusing to fix bugs at any cost.  If you're a non-profit exec, these all cost you a lot of money.

If the person asking does community organizing or grassroots fundraising, CiviCRM is the only best-in-class CRM geared specifically toward those tasks.

If they're using Excel right now, it's the ability to get all their data in one place - not in someone's phone (or head), not spread across ten spreadsheets.  Something as simple as "I want to send an e-mail to everyone we work with in San Diego" is a very difficult task without a CRM!