Many ArcGIS Online maps utilize basemaps not hosted by ESRI. If the provider of a basemap takes their server down (for maintenance etc), any ArcGIS Online map that utilizes that basemap will be unable to load. I propose that we be allowed to specify an alternative / fail-safe basemap that the map would attempt to load in the case that it was unable to load the primary basemap.
In our particular case, we utilize imagery provided by our state data center for the basemap on most of our AGO maps. When they take their server down for maintenance, our maps go down. If the map were, instead, able to load an alternative basemap (ESRI, self-hosted, etc), then the maps would remain live until the state's server is put back online.
It could be implemented by adding a "Specify Alternative Basemap" to the options popup when you click on the Basemap in any given ArcGIS Online map.