Twilight is a bit of a sleeper problem. On one hand, its hard to point to specific, traditional problems that media might have, I.E. Violence, blood, gore, sex, etc. The quality of writing also clearly skews the entire series towards the, "Guilty pleasure" end of the spectrum, trying to be a fun, cheesy, escapist romance. Certainly I won't judge anyone for enjoying this guilty pleasure. The problem this franchise presents is 2 pronged: first from the subtle messages, and the second from the fan community.
First, it's worth looking at the message. Certainly there isn't anything to point to that is outright bad on its own, but the far more meaningful, instructive message underneath needs to be addressed. It's far to easy to look at Twilight as an instruction manual for an abusive, codependent relationship. Note how Bella, as a character, seems to have absolutely no personality outside of being defined by the guy she likes. The guy also comes across as a bit of a creepy, controlling stalker. A lot of the tense, dramatic moments fueled by the condition of Edwards vampirism can easily be taken by younger children as a justification for situations where someone needs to call the cops. If a child is at all likely to model there views of relationships off of what they read, then seriously consider not allowing access to this, or at least having a serious talk about the separation of reality and fantasy.
The second issue, and perhaps a slightly unfair one, is the fan community. Certainly there are plenty of normal Twilight fans, and most popular intellectual properties have a seedy underbelly. However, Twilight's bears particular concern. A cursory Google search will be filled with disturbing instances, up to and including acts of vandalism, assault, and threats, all based on opinions regarding the book or the characters. Even the Actor who plays the male lead has stated concern regarding fanatic fans who cut themselves, and try to get him to drink there blood. Certainly you can't judge an individual based on there tastes, but be aware that this underbelly exists, and it is not a positive thing.
Overall, it is easy to miss a need to remove Twilight from your child's media diet. Its influences are more subtle then can be presented on a simple G-R scale. I would recommend going a little deeper, reading plot synapses, and critical articles on the franchise before making a decision.
In the interest of full disclosure, I personally can't stand Twilight, and personally judge its quality to be very, very poor, regardless of its adult content. That could skew my perception. Take of this what you will.