The proper nomenclature is to use single quotes, but you're apparently referring to the "cultivar epithet", also sometimes referred-to as the "clonal name", which is used to distinguish one plant from another of the same breeding.
For example, let's say you had a particularly nice Dendrobium nobile (it's a species, so the specific name is lower case), so to separate it from other D. nobile plants, you can dub it "Harry" or whatever unique name you wish by attaching that in single quotes: Dendrobium nobile 'Harry'. Going forth, any division or clone of that plant (asexual reproduction) would carry that same designation, and cannot be ethically changed.
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