Generally, the autonym is probably more useful when showing a language list. This has been the practice in Wikipedia since interlanguage started, and this is the practice in many websites. It makes sense: for example, when an English speaker is looking at a Japanese website a label that says "English" is more useful than "英語".

However, there are also scenarios in which showing the name in the language of the user or of the wiki. There are gadgets that do this for the interlanguage links list in English, Portuguese, Russian, and other Wikipedias, so obviously it's useful for some editors. This is also the practice in the Wikipedia iOS app (though not Android for some reason): the autonyms are shown in larger letters, and the languages names in the user's language are shown in smaller and faded letters.

In ULS (at least Compact Language Links) the language name in the user's language is shown in the tooltip, although it's probably not a perfect solution.

Some user scenarios in which this can useful:

• Accessibility: Screen readers for visually impaired people don't necessarily support other languages. It's more reliable that a screen reader will be able to read in the language of the user. If a screen reader doesn't support a language, it may read out the letters as Unicode character numbers, which is rather inconvenient. (This was found by @dchen as part of T183175).
• Many monolingual editors said that they want to look at articles in other languages even if they don't really know that language. This is useful, for example, for:
  • looking at an article in the language that is related to the article's topic (for example, a French article about a French city)
  • finding images
  • comparing article structure and size between languages
  • for bolder editors: improving articles in other languages, for example adding number values to infoboxes

However, Wikipedias is some languages have gadgets that show the language names. For example, the English Wikipedia has "SidebarTranslate" and the Portuguese Wikipedia has "Tradução Interwiki", and they are enabled by many editors, so this must be useful for at least some people. Similar requests were also made by the participants of the discussion about Compact Language Links in the German Wikipedia.

Making it available to more people as a proper MediaWiki feature rather than a gadget is worth exploring, but it will require proper design work, with definition of audience, testing, and so on. This is just a proposal.

This is an exploration task.