In the Item details page of a web map that I am owner of (also administrator) I select "edit" and re type a new name. When I push save - the thumbnail diappears and a "please wait" message appears and the application hangs.
James, your maps are not visible to us. You would need to share them with everyone.
Nicholas, Your web map seems to reference map services that are on an internal network, so we can't see the layers in them. Can you consistently reproduce this? If you can document exactly what you do we can try it out.
We are experiencing something similar with our subscription account. Over the past two weeks we've had a lot of difficulty editing the ArcGIS Online properties of web maps or apps. For example, even though I am the site admin and owner of the particular maps and apps in question I can no longer assign thumbnails in the edit properties dialog. We are also getting occasional errors when we assign these web maps or apps to the group that feeds our modified Public Map Template - the link URL that is passed to the template is incorrect.
These problems all seemed to have cropped up with the December release.
Unfortunately I can't expose these apps for you to test since they sit behind our protected domain. Regardless, something clearly broke with the December release.