There's disagreement about this. Many PTOs have membership requirements similar to those your group is considering, especially dues in exchange for voting rights. But I think such requirements do act as a deterrent, and in the case of your PTO it seems they're being considered specifically to deter one person. That's a dangerous message to send to the newbie and other potential volunteers.
There could be legitimate reasons for alarm over this person's sudden candidacy -- but they can be dealt with by giving all candidates time to "campaign", to state their reasons for wanting to be president, their goals, and so on, and having a vote. That's what elections are for! The current president would be able to explain why experience is valuable and express concerns over the newbie's lack of it, while the newbie would be able to explain why she'd be great anyway.
As you say, it's a great problem to have. But I would be more concerned about keeping everyone -- your current officers and the new candidate -- involved and engaged regardless of the outcome of the election than about writing rules to stop new people from running for office.
We have had a unique issue arise this year...we had a new (to our meetings / not to the school) parent show up at the first meeting and throw her hat into the ring nominating herself for president (which is a great problem to have). The officers and usual attendee's were shocked by a new person volunteering for such a position...typically new attendee's like to work their way up the "officer chain" by starting out as a chairperson of an event and learning the ropes so to speak before jumping into such a role.
Needless to say, it has put the officers on alert - wanting to define PTO "member". They would like to see members attend a specific number of meetings and be involved in planning & volunteering at 2 PTO sponsored events during a calendar year to be qualified as a member...thus giving you voting rights. Attendee's could weigh in on the topics but only members would be able to vote. In addition you may only be elected to office if you have been a regular member (unless no one qualified wants the position).
I'm a worried that defining members will be a deterrent....Thoughts?