Thanks Derek. You were right, I was missing the "Apply" in the web part properties. I was looking for a way to save my changes to the map from within the web part.....similar to where you'd save when editing an ArcGIS online web app.
Just so others are clear that read this thread, the steps I need to take for adding a data layer to the ArcGIS maps for SharePoint...and saving it... are;
1) Start editing the page
2) Start editing the web part....to turn on the ArcGIS Online Map Content properties dialog box
3) Add data to the Map content
4) Click "Apply" in the ArcGIS Online Map Content properties dialog box
5) Click "OK" in the same box
6) Click "Save" in the ribbon of the "Page" tab
7) Lastly, stop editing the page