The Basemap Gallery should be customizable. Currently it is customizable only for all users in an organization. It would be much better if the individual user could customize the basemap gallery so basemaps they use frequently are easily available. This should be customizable at the web map level. For example, suppose I want to use a particular basemap on a web map. That's fine until the user decides they want to see a different basemap for a moment. Then the user will not have the option to get back the original basemap.
It would be even better if the basemap gallery could be customized by user and by group, as well. I would envision the gallery might be divided into sections including the ESRI standard set of basemaps, any custom basemaps the map author added to the map, and any custom basemaps the user and/or group has defined.