OK, using the RESTClient (https://addons.mozilla.org/en-US/firefox/addon/restclient/) add-on in Firefox I was able to succeed in doing this. The add-on gives lots of info about the request.
Here's a link to the doc that helped:
The critical change was to add double quotes (") around the uri of the member to be added:
to do this in C# means using extra double quote characters...
string value = @"[""" + ConfigurationManager.AppSettings["uri_prefix"] + "people/" + memberID + @"""]";