Why can’t I edit a field?

That padlock means it's a locked field. Whatever created it told Jira not to let anyone reconfigure it, so the usual path to add options (Issues > Custom fields > your field > Contexts and default value > Edit options) is greyed out or missing entirely.

Depends what owns it. If a Marketplace app created the field, that app controls the option list, so you'd add the new values inside the app's own config. Not in Jira's field screen. If Atlassian locked it (a system-managed field), you can't extend it at all, and rebuilding it as a new field is the way out. Same annoying outcome you're trying to avoid.

Can you check which it is? In Issues > Custom fields, find the field and see whether it says it's managed by an app, and which one. That decides whether you add your option in the app or have to rebuild it as a plain Select list (multiple choices) field you control.

Atlassian's page on this is titled the same as your question, by the way: what are locked custom fields.

Hello  @Brett Denison 

It depends on the type of field you are using and from where it coming from

If It was created inside a team-managed space, In that case, it may appear locked in the global Jira field administration, but its options must be edited inside the space:

If it is a global/system field, some of them may be locked and cannot be changed freely. But if it is a custom field created by you, you should normally be able to add options through the field context settings under  Jira settings → Work items → Fields → Contexts and default value → Edit options

Best,

Arek🤠