On beta sites, the privilege to close questions requires only 500 reputation. On other sites, this privilege requires 3000 reputation.
On Board & Card Games, I have over 500 rep and can close questions - if I'm using the website.
On the Android App, however, this doesn't show up in the question options. It's usually the last item on the list.
I'm using Android App 1.0.26 on a HTC One running Android 4.4.2.