Let's take a brief look into possible missions for a testing dojo and give some recommendations. You can find a selection of missions on Testing Challenges.org.
Test this
The classic mission involves testing an application. The variety of applications includes open source programs, commercial software available in your organization or even your company's latest product. Such a session can end up as a bugfest. The facilitator needs to check if the mission is doable before the session starts. The company's firewall may block some content on a web page or it might be cumbersome to test a web page with unstable network connections.
This type of missions has many different forms. The facilitator can focus the session on a particular aspect of the application like usability problems. The audience could get to know how to learn and model a new product without thorough documentation. You could also pick test automation, though this needs more planning and preparation from the facilitator's point of view. The facilitator should provide some initial examples of automated tests and then ask the participants to extend them. Usually during a short period of a single hour or two, automating test cases for a new application seems to be not as productive as an exploration session conducted with mostly manual tests.
Evaluate tools
A mission to evaluate a tool could use mindmaps for test ideas or try out a particular test tool for the whole session. In such a mission, the product under test is usually the tool itself, but you can run it also for a common program that you test at work and compare the results directly with your daily work. Evaluating tools can help to decide if you want to learn more about them.
You could evaluate an open source application for writing test documentation similar to the company standards or assess several free online services such as URL shorteners to be incorporated into the next product. Yet another mission could be to evaluate a new bug tracking system and provide management with information regarding the capabilities of the tool given the company's context and culture.
Learn new approaches
There are many testing approaches to try out. You could focus the mission on some particular mnemonic like FCC CUTS VIDS or use soap operas to generate test ideas. Like tools evaluations, this type of mission aims to try out and learn about new approaches. After these sessions, the whole team will have made some experience and can make a more informed decision about the usefulness of the approach.