I ended up going down a bit of a rabbit hole about the etymology of Aden, and my findings are that there are competing claims about the origin of the name. (Note: I do not speak Arabic nor do I have an academic background in Semitic languages)

Theory 1: Sumerian eden "plain, desert, steppe, open country" → Akkadian idīnum "desert, steppe, plain" → Arabic ʕadan "Aden"

Theory 2: Akkadian adānum "fixed date, time limit, period of time" → Aramaic ʿeddānā "moment, season, age, time; (Hatra dialect) halting place, station, caravanserai" → dialectal Arabic ʕadān "particular time when something is set; place of remaining, spot where one bides; bank, shore, where water meets land, water end of a journey" → Arabic ʕadan "Aden" (either a direct descendant or a doublet).

ʕadān is also where you get ʕadana "to reside in place; to stay, bide, remain; (with al-bilad) to settle a country, region" from, which is in Arabic script^[I opted to omit Arabic script from this comment because the Unicode bidirectional text kept breaking on me, curses.] spelled the same as ʕadan save for the diacritics, hence this map's claim that that's what Aden literally means. But it seems like the actual etymological meaning of Aden (in theory 2) is probably something closer to "the station, the inns, the pit stop, the rest area", alternatively "the end of the line", alternatively alternatively and most boringly just "the settlement" (Christmas Island sends its regards). The through-line in any case seems to be "this is a port where you can rest between your long voyage at sea and your long voyage over land".

Also, Sumerian eden and the unreconstructed proto-Semitic ancestor of Akkadian adānum have both been proposed as etyma of Eden of "garden of" fame, which means that there's a good chance Aden is cognate with Eden either way.

You should watch SPARKLE ON RAVEN: The Life of DrillGirl; it is a very good show. {Cēterum|Furthermore} {cēnseō|I think} {iūs auctōris|copyright} {esse dēlendum|must be destroyed}.