Great question. A lot of this has to do with the type of role you are interested in. There are vast differences between hiring someone to fill a technical role than a business or product role. Given that, there are also similarities that all successful startup employees have.
At my company we really like to find people with an entrepreneurial spirit. We realize as a startup we are not going to keep people employed here forever. We are but one stop on a long list of companies that a person will work at in their careers. Having that entrepreneurial sprit is obvious in those that are driven to succeed, are willing to wear many hats, and push past the "no, that can't be done" attitudes. It takes a lot of strength to be able to start a company up from scratch and having these qualities at any level of the organization is imperative to success.
On top of that, it is important that startup employees are constantly looking to learn new things (deepen your understanding of a market, trying out new technologies, learning other aspects of the business like finance or marketing). This will increase the value you are able to give to the startup but also proves great as a way to increase the value you take away from that startup.