Is there any biblical support for or against a woman president?

I see no problem with a woman president. For one, Deborah was a leader in the Old Testament. For another, the Old Testament mixes national leadership with theocracy, so we might not expect to find relevant examples here for us today. Finally, when it comes to the explicit restrictions on the role of women (esp. in 1 Tim 2:12), the nature and scope of the restriction has to do with communicating spiritual truth among a body of believers.

As to whether a woman would make a good president, her merits must be decided on entirely other grounds and without influence from this issue.