well, for starters, you could look up to http://solutions.arcgis.com/ and search for specific ArcGIS-related projects for various purposes for ideas. Also you could browse for implemented ideas on arcgis online (and other ESRI projects) in https://blogs.esri.com/esri/arcgis/.
I would personally suggest you to develop some kind of social media related mapping project (i.e. crowdsourced map). This could produce many kinds of information system (for example, this project which utilize the social media for emergency and disaster management). Given correct keywords, you could adopt the idea into various purposes, such as sentiment analysis and mapping, geopolitics, traffic monitoring, etc. The limit is your imagination