To some extent, the way you reference archival material is affected by what you are using it for. It is a question of balancing internal consistency with the required style, etc.
It is important to take into account how the museum/archive in which the item is held wants its material to be referenced, so it is important to find that out when consulting the item. This will include the correct form of wording for the name of the archive and the way to give the reference number for the document or item.
For publication
Publishers will have their own house style and protocols, and if publishing, for example, a photo of an object or document, you would need to say ‘used by permission …’ and to use the format the owner or curator has given you.
For thesis bibliography/referencing
Follow the Faculty requirements (e.g. Harvard) and University guidelines as closely as possible and come to an agreement between supervisor and student regarding how archival resources will be referenced. Archive material is usually listed separately from published works in the bibliography, in a section called ‘Archival resources’.
Cite Them Right has a section for manuscripts in Harvard, which is useful as a guide.
The key elements which must be included (this is for archival material, slightly different to museum pieces, which often have their own accession numbers) are:
In other words, the information needed to allow a reader to find the document themselves.
If the record/document has been accessed online, you should also provide a URL and the date of access.
The National Archives provides a guide that is mostly relevant to their own material, but Section 11 has some information on referencing material in other archives. This page from Archives Hub may also be useful.
Examples of standard formats