Nobody wants to be President

Hog tie them and shove them into a flammable chair. Give out the pitchforks accordingly.

No one wants to do it. I suggest creating a healthy team environment where one would feel okay with being in that position.

Ask what it is about the board position they don't like. They may have some misconceptions about what it actually involves. Clear those up and see if they change their mind.

I'll bet anything your board member is balking at running the meetings or serving as "the face" of the board. Delegating these tasks may work (see if your bylaws allow it). Signing contracts seldom causes much angst (can probably also be done by the secretary) nor does serving as liaison with vendors or others (also can be done by other officers).

Delegating can work for your benefit. For instance, I'd rather have a root canal than run meetings. But we also employ a community manager and one of the things that person can do is run the meetings. I happily left that task to her while I did the draft minutes. We had the more skilled persons handling these tasks, and I got to focus on what was being said rather than the mechanics of the meeting.

Let your introverts introvert, and put the more extroverted types in roles where they shine.

In our Board (CA), the 3 positions are created equally, no special responsibilities and our PM runs the meeting, which often seems like NASCAR!