I ran into the same problem, and here is what finally worked.
The shapefile that ArcGIS needs include the .dbf, .prj, .shp, and .shx files. When I had created a shapefile from another program, I also got those four, plus a .cpg file. And I couldn't load the five files in a zipped folder--I kept getting an error. What finally worked for me was making sure the name of the files have no spaces. Then, I erased the .cpg file, placing only the .dbf, .prj, .shp, and .shx files into their own folder for the layer. Then, I zipped that folder, which was about 6MB. Finally, I dragged and dropped that zipped folder into the prompt box that you get when you add a layer from your own files into ArcGIS online.