There are two sides to this question:
Side 1:
How do I remove a specific build from the build history (not just "old builds" or "builds that meet parameter X") using the GUI? The reason I ask for the GUI is because I don't know how to use Jenkins from the command line, otherwise I'd try this SO answer (tangent: tutorial for Jenkins CLI?).
Side 2:
How can I run a build such that it doesn't appear in the build history? A 'debugging' build?