My initial reaction to your question was, This smells a lot like what Jakob Nielsen calls "mystery meat" to me, but it is not immediately obvious why. And I don't believe that's a good enough reason not to do it.
I think there are two cases that you need to consider, and some caveats. Rather than give you my subjective opinion, why don't I give you some pointers with which you can decide for yourself.
UI interfaces do exist with "user editable" content. Windows, OS X, Linux, iOS, Android, etc. It doesn't matter which you choose; they all allow for customization of menus and widgets. So we know that the paradigm exists.
Consistency is king in all things. When it comes to web sites, people tend to like when things stay the same, as expected, as remembered from their last visit. The web is not a supermarket and you can't move things around hoping that they'll discover something else they also like on their "journey". That said, if your site is more of a functional application than a destination, then I believe you can get away with (nay, enhance with the notion of personalization) this concept. Plus it adds an element of cool and says, "we care that you own this".
However, we also have the problem of "remembering" these settings. Will it done by user account control? What happens when they're not logged in? Is it okay that these menus will be "default" in that case?
And lastly... Does moving, adding/deleting menu items add anything concrete to the user experience? Are there items there that some absolutely will not use? Is it possible to revert back easily in the case of an accidental removal, say?
Hopefully, that gives you some food for thought.