Seeing the Aurora in Iceland is a little unpredictable given clouds. But on clear nights when the Aurora is reasonably strong, you can even see it from the periphery of Reykjavik
I mostly used AuroraForecast.is
Two very good options near Reykjavik are:
Seltjanarnes via Bus 107: A separate town on the northwest tip of the Reykjavik peninsula. Take the 107 to the end of the line and walk to the Grótta Island Lighthouse
Alftanes Beach via Bus 23: Alftanes is a peninsula just west of Reykjavik. The official residence of the Icelandic President is nearby.