Warning: Event layer not editable
Event layers that are based on tables without ObjectID fields cannot be edited in ArcGIS.
Solution: Export the event layer to a new feature class or use an editable table for the data source of the event layer.
More information: Although you are unable to edit this table in ArcMap, you can edit the table the layer is based on directly on disk, such as in a text editor, if the file is a .txt file—including changing the x,y coordinates of the point features. Your changes will then be reflected on the map next time you refresh the map.
However, you can start an edit session on an XY event layer if the table it is based on has an ObjectID field and if the data source is editable. This will let you edit the layer's attributes in the table window including editing the x,y coordinate fields manually to change the location of the points in the layer.
Exporting the event layer to a new feature class, such as a shapefile or geodatabase feature class, allows you to edit the features fully.