Only the author of the map notes layer can edit and save the layer in ArcGIS Online Map Viewer Classic. To allow other users to edit the map notes layer, publish the layer as a hosted feature layer. Refer to FAQ: Is it possible to export map notes from ArcGIS Online for instructions. After the hosted feature layer is published, click the More Options button and click Open in Map Viewer Classic with full editing control to open the hosted feature layer with full editing capabilities. This option allows editing the values in the attribute table, as well as editing the symbology.
Note: Only members of the shared group and members with sufficient privileges have access to edit the hosted feature layer with full editing control. Refer to ArcGIS Online: Add layer to new map with full editing control for more information.
Alternatively, ArcGIS Online Map Viewer also allows editing a map notes layer as a hosted feature layer. Refer to Problem: Unable to edit map notes layer in ArcGIS Online Map Viewer for instructions.