Got it - since we don't have an attendee-level PUT API for this kind of data, I recommend making a single profile question for the entire org, then overriding it for each event. Once you have the question created you can call the PUT request on the members API to save the value into that question each time.
To see the data structure for a member, call the GET /rest/members/{member_id} API with the ?fields=questions parameter and look at the data coming back.
When you're ready to update the question, you'll send back a slimmed down member object in the PUT request data with a 'questions' object containing one question: the question UUID in 'id' and the value in 'answer'.
For the member ID in the URL, you easily can get that from the URL in the People screen when you click on a specific member (good for testing on a specific member like yourself). You also get back the member ID when you call the attendees API as part of the 'profileuri' field.