Definitely lease, if they can't afford 100% upfront payment.
Cause I am wondering, in the case of equity partial ownership, how are you really gonna determine your annual income (assuming there will be any)? The company where no one knows where they are, what they do, how they work, what is their field, ground, financial situation, plans, ethics, etc? They can draw any reports they want.
Also, thinking of startups, people automatically think about Facebook or Twitter, but for one FB there are 10000 startups that go bust in 3 years, or stay low-profile/low income for decades... And even imagine for a second it's ''new Facebook'' - but you do remember Eduardo Saberin vs FB case? If not, the ''Social Network'' is a movie to watch.
Bottom line, my point is that the idea of partial ownership of a share in exchange for domain name will result in a wasted domain name, nothing else.