For the last few years Verizon has hosted the Innovative App Challenge for students. Verizon recently published the list of the best apps developed in the 2015-16 Innovative App Challenge. The top app was created by middle school students in Michigan. Their app provides a way for the public to alert local government to safety concerns in public parks. The whole list of top student-created apps can be seen in this PDF.
The next Verizon Innovative App Challenge begins in August. Students must make a video to explain the concept of their apps and write a short essay about their apps. Complete entry rules are available here.
If your students do want to build an app either for this contest or for their own enjoyment, the MIT App Inventor offers a great way to do that. Students can build working apps in their web browsers before using them on their Android devices. Click here to read more about the MIT App Inventor.