Looks like an identified issue since 2009 based on this thwack post:
https://thwack.solarwinds.com/t5/SAM-Discussions/Can-t-rename-first-component-monitor/m-p/291570
Still the default behavior now. It would be handy to have the first one renamable as well so I don't have to keep making the first component be an empty monitor so I can have the rest named as needed.
You can always rename any component, you just have to rename them at the template not on an assigned instance. The scenario you are probably running into is a template with a single monitor, like the OOTB http template, where you have added additional components directly on that instance so they don't exist in the template.