Rido, your code is really hard to read for me because of the way it's mixing together a few things and not following any of the usual conventions for Backbone.
For your specific problem, I have a feeling the problem is that you binding both the Edit view and the New view to body (el: body), both respond to the event submit, and you are never clearly cleaning up the views, so I think that whenever you add a project and then edit it, the add view is still in memory, still bound to the submit event and still answering the call = new project with the new name, instead of editing.
It's 'easy' to fix in a dirty way, by adding a call to stopListening after adding, but the real problem is that you are binding to body, and mixing togethers the Backbone Router and manual hash control + other weird patterns, such as fetching the collection every 5 lines (you could just create one at the start of the App and always use it! here it's localstorage so it doesn't matter but if you ever move to a remote storage, you'll regret it... fetch() reset the collection and do a full reload!). May I suggest you maybe try to rewrite this without caring about jQuery mobile and just try to make it work with Backbone.Router + a single collection + not binding to body but instead, create the views on the fly and append them to body / remove when you are done? You'll see the bugs will be less weird and easier to track.