Happy to share the typical things I've done on Wikidata that are not "Wikidata Game" or "Mix-n-match" unless you want that as part of your study too.
In general, I have worked on cleaning up the firearms ontology, moving things from instances to subclasses, and then scraping categories from en.wp using Petscan and massively putting them in the right subclasses in Wikidata, and then adding English labels to items that exist in other languages only.
As for tools, I also have developed a script for Google Spreadsheets that works in conjunction with the Wikipedia and Wikidata Tools for Google Spreadsheets, so that I can get a high level view of certain subjects and their article quality in EN.WP, and see what the Wikidata coverage looks like. I've shared this script with Women in Red (@Rosiestep), Women Scientists (@Keilana), Library user groups (@Gamaliel) and also the Auckland War Memorial Museum Wikimedian in Residence to look at sets of articles to assess quality, and then generate Quickstatements to help fill in descriptions, or other properties that need work. I've demo'ed this at Wikimania recently in a Lightning Talk:
I also use ORES to evaluate article quality as part of this, but that goes beyond the Wikidata interest. :)