A number of travellers come to Iceland for one reason: The northern lights. People probably don´t know that the northern lights are swirling above our heads all year, but in summer we just don´t see them because of the long and bright nights. You need darkness in order to see them.
Soon the organized northern lights tours will start from Reykjavík and other towns and places in Iceland. Sometimes you don´t have to go far from Reykjavík in order to enjoy the spectacle, just a 15 minutes drive, and sometimes you have to drive 1-2 hours. All depends on the clouds and you can see the prognose for the sky on www.vedur.is.
If you want to travel away from all lights then a tour to the highlands might be an option: https://glacialexperience.is/destination/wonders-of-the-highlands/