How do I configure a free event ticket option for internal use without it showing up on the website?
If you only want to have the free registration be available for back-end registrations you should create a price set. The options for admin-only registration types should be in their own price set field, since only price set fields can be marked "Admin only," not an option within a price set field.

Alternatively, if it's just a matter of the participant not paying and you don't have to report on who's a paid participant and who's a comp, you could just register them in the back end and not record a payment. You could still make this work without a price set by creating another participant type of "Complimentary."