First. I would be sure I had all that was needed and approved with a full description/spec of the changes. I would research any new technology requirements, depending on how much was involved. I would be sure to make a back-up of the full site and database. If possible, I would make the changes to a mirror site on the same server. If this was just an edit or text change I would create a copy of the page being edited and save it with a different name. Make the change. Verify the change was correct and then publish/overwrite the original page.
If the change was more involved like dynamic or a script that was being used for the first time. I would have all the changes done on the mirror site. Then tested and approved. The changes could entail the work being done by many departments and could quite possibly involve a project manager. Each stage of the changes would need to be tracked and verified prior to moving to the next stage/department.