You can add common names on the taxon page under the ‘Taxonomy’ header:
From the following page you can enter the desired common name and associate it with a language on iNaturalist:
After that you can also choose to have it associated with a particular locality in the event that there are multiple common names in the same language and only some are used in a geographic area. (Note that I believe this is only possible if you are entering the common name yourself, otherwise you can contact a curator to have them make the change for you.)
Additional information courtesy of @cmcheatle: