At least something has been named after Genghis?
The largest airport in Mongolia is Chinggis Khaan international airport.
I am suggesting geographical features / costumes / customs / architecture / laws - anything.
From Wikipedia: Yassa (alternatively: Yasa, Yasaq, Jazag, Zasag, Mongolian: Их засаг, Yehe Zasag) was a secret written code of law created by Genghis Khan. The word Yassa translates into "order" or "decree". It was the de facto law of the Mongol Empire even though the "law" was kept secret and never made public. The Yassa seem to have its origin as decrees issues in wartime. Later these decrees were codified and expanded to include cultural and life-style conventions. By keeping the Yassa secret, the decrees could be modified and used selectively. It is believed that the Yassa was supervised by Genghis Khan himself and his stepbrother Shihihutag who was then high judge (in Mongolian: улсын их заргач) of Mongol Empire.
Asians are(were?) known as mongoloids in science.
Asian babies are born with bruise-like blue spots that eventually disappear called "mongolian spots".
The Chagatai language, which was named after Chinggis Khan´s second son was widely spoken in Central Asia and was an important Lingua Franca.