*edit* New thought: Maybe they can introduce Rand and have a professional rivalry going on between the two. Not a romantic rivalry. Maybe like Rand is the rising star in Uhura's department and they are constantly trying to best each other. Could drag this out over the next few movies.
I'd actually prefer Rand to stick with being Kirk's aide (sort of a security liasion mixed with a secretary) as it gives her an opportunity to hang around with him and get in on the action. Linking her to Uhura's department ties her to a console and we know from poor Uhura and Chapel how being tied to your post inhibits any character development.
I agree that they should avoid romantic rivalry beyond perhaps the odd suggestive glance from Rand. One crew relationship is enough for a two hour action film and Kirk needs to stay free enough to meet Carol!
To use Uhura more fully they have to get her off the ship.