Great idea! National University of Singapore has created an anonymized "data lake" that researchers can apply to work with: http://nus.edu.sg/alset/data-lake.html
They also have a synthetic data set generated from their data lake that can be used by anyone, including students. This kind of resource could be very helpful as a learning tool at your institutions.
I think there is room for a wide variety of student projects in learning analytics, from technical coding (e.g. implementing a new algorithm in Moodle learning analytics) to the humanities (e.g. conducting qualitative research to study the understanding, attitudes and concerns of students, faculty, and others regarding the use of learning analytics).