>Is there any official procedure for giving an ancient site a name, or renaming?

No.

Sites usually have a traditional name, given by the locals. If not, then the name of the location is given.

Who first gave the Holme Timber Circle the inappropriate name 'Seahenge'? Probably an ignorant journalist.

Sites are given the name that they are called. If we all call 'Circle 275' Circle 275, then that's its name. If we all call it 'Bob' then that's its name.