There is a certain appeal to Robin finally being Batman's son, isn't there?
I suppose at this point in time it makes more sense for Tim to take a new name and Damien keep Robin indefinitely. Actually, Tim needs new name no matter what, because Red Robin is about the worst name this side of the Whizzer.
But ten years ago, or whenever it is that Damien took the mantle? I think he should've taken something original instead. It also feels fitting for his personality; would Damien ever accept a hand-me-down codename held by (to his mind) inferior people?
But at this point, it's Tim who needs to find a new name, and Damien should probably hold onto Robin.