Ultimately, the profile you're looking for is the Old Warhorse who's worked in and around the same industry/niche you're targeting. They want a piece of the action, willing to invest and provide advice. And importantly, trust you to run the day-to-day. Look for former senior operators/builders with deep personal networks.
I currently sit on the board of my community association, and enjoy the advisory style of influence. I'd like to continue this experience with startups, non-profits, or for-profits.
I'm early in my career, and it sounds like a lot of the qualifications are time, experience, network, or money - but I'd like to take any active steps that I can at this time.