There is no REST API call for this as far as I know.
A plugin can probably do it. There's this class : http://docs.atlassian.com/atlassian-bamboo/latest/com/atlassian/bamboo/configuration/variable/ConfigureGlobalVariables.html but not sure how it's intended to be used. I would imagine you probably want to alter a global variable during a build (as part of a Task in a Job? If so there might be permissions issue since a Task is usable by any usable and altering global variables affects all users, so it might be limited to system admin permissions.
Hopefully one of the Atlassian folks can chime in with a better answer...
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