The name choice is completely up to you. There are PTOs and PTSOs. Just as there are PTAs and PTSAs.
After you pick the name, GJ is right that it's then up to you whether and how students actually have a voting role. The "S" in the name, if you add it, doesn't definitively mean that students vote. That woud be defined in your bylaws.
I don't think they'd necessarily have to have a vote; your by-laws could make them non-voting members, just as some PTOs exclude Ts from taking board positions. I'd be concerned that a group of kids might drop in on a lark and vote for something bizarre. If you want student input, then you shouldn't wait for someone to ask to stay (and the name change won't affect that).
Hi - we are forming a new organization, spinning off from PTA but I don't see much about the "S" for students with most groups being PTO's. Not that it's a big deal, but I need to make sure the name is the same for the EIN and 1023 Tax Exempt appl.
Is there any reason not to include the "S" so we do not limit input from our students. We have worked together with student council who reports at each (PTA) meeting and would like that to continue.
I understand they would have a "vote" at a meeting, but for middle school that might be ok - although no student has ever asked to stay and vote at a meeting.