What productive academic work can you do with minimal attention in a small (<30 minutes) block of time?
Given that you can't do something that requires your full attention during this time, I would at least advise to follow Matt Might's advice to procrastinate productively on meta-work:


Read articles (or answers on Academia.SE!) about how to be more productive as a grad student.
Read a few pages of a book on data analysis, visualization, academic writing, or some other research skill.
Use the time to review your schedule, make to-do lists, schedule a meeting, etc.
Try a relevant Coursera course that has video lectures broken into small bits
Put some thought into a tiny bite-sized aspect of your research or research presentation: How should I organize this manuscript? What is the best way to visualize this data?
Unfocused literature search: check in on your favorite journals or conferences and identify interesting papers (to read later, when you can give them your full attention)
Do a favor for someone else: Even if you're not teaching or grading this semester, I'm sure someone in your department is, and would be happy to give you a pile of quizzes to grade.
Update your CV and/or web page.
Start preparing a presentation on your current work. You'll present it to someone, somewhere sooner or later, right?
Think about your ultimate career goals.
Use the Internet to look for promising potential collaborators at other institutions.
Try out a new programming trick, LaTeX package, or software tool.