This can't be you. Why not? Because to get the best out of your board, the directors or advisors need to feel able to challenge and question you. That's a lot less likely to happen when you run the meeting. What you want is someone you trust, preferably someone you admire and who has the success of the business as his or her only goal. This is a mentoring role, not a career steppingstone.
The best chairman I've ever worked with was excellent at silence. He managed discussions very thoroughly but largely kept himself out of them. He made sure that no one was left out and he ensured that the questions raised by advisors got an answer. This didn't mean that he kept himself out of the debates. But he listened as hard for what wasn't being said as he did to what was. He was also not afraid to make an argument that was counter or oblique to the main trend. People quickly came to trust him because he stopped the board from being too conformist. This is a rare and great gift. If everyone is thinking the same way, you aren't getting the best from the talent you've assembled.
Many people imagine that a good board is one in which everyone gets along and there isn't any conflict. This is true for publicly-traded companies too. In both cases, they're wrong. Conflict is how thinking happens in groups. No conflict, no thought. What you want in your chairman is someone who isn't afraid of conflict and knows how to do it well.
Tact is also critical. Much that is discussed in board meetings can't or shouldn't be aired in public. Some directors are more sensitive to this than others but everyone should take the lead from the chairman. If you are very lucky, you will find a chairman who also understands that you want to have conversations privately about matters that don't get aired with the board. This can, and should, be a rich mentoring relationship if you both know how to use it wisely.
What that implies, of course, is that chairing a board of any kind is time-consuming. So you want to ensure that your chairwoman understands the commitment and is prepared to put the time in. I've often been asked to chair the boards of companies and organizations I was happy to advise, but knew I didn't have the time to for it. This is a critical difference and you want to be sure your candidate fully appreciates what's involved.
Should you pay your chairman? The time and responsibility involved in chairing any organization means that you need to show your gratitude. Typically this means reimbursing all reasonable expenses and offering an honorarium or shares in your business, depending on your ownership structure.
The best chairmen and chairwomen I've ever seen have felt that what they invested in the organization wasn't just their time but their reputation. They sought the company's success as avidly as any founder or employee. As such, they bring extra leadership and dynamism to the company as a whole because the person they chiefly inspire is you.