TL;DR
If your employee truly is bored, and one or both of you think this is actually a problem that needs to be solved, ask her for solutions that suit the both of you. The path to professional self-actualization for both of you starts with communication, collaboration, and alignment. You need to address those dynamics before you can properly measure or incentivize a potential solution.
Analysis
I know the Junior PM is getting bored because they haven't been super busy.
This looks like an assumption, not a self-reported fact from the junior project manager. Make sure you aren't creating problems when there isn't one.
I am wondering if anyone has developed any small projects for their team to work on during a down time?
This is an X/Y solution that also runs afoul of the 100% utilization fallacy. Top-down busy work is never a good solution for issues of morale or employee engagement.
Recommendations
The "solution" isn't to find small projects to keep people busy. The real solution is to use this time to improve your communications with this empoyee, and to find opportunities for continuous learning and self-directed engagement for both of you.
Start with communications. Knowledge workers who feel engaged at work don't feel "bored," and more importantly engaged people don't feel so bored that they might quit. So, be sure you ask your employees how they feel about their roles, and about their level of satisfaction with the work they're doing. Job satisfaction is only a problem if the employee thinks it's a problem, so don't make assumptions about how people feel or what brings them job satisfaction. When in doubt, ask!
Next, assuming you think there's a problem that needs to be addressed, address the root cause and not the surface issue. For example, if you are concerned about losing a key resource, or about lack of professional development, then address those specific concerns. On-the-job or cross-functional training might be useful, but the goal should be to learn rather than to keep busy. Partnering with other project managers can often provide organizational support, learning opportunities, and esprit de corps without creating make-work.
Most importantly, ask questions and solicit solutions, rather than "solving" the problem for others. As a leader, your goal should be to cultivate self-directed engagement in your staff. You do that by empowering people, creating opportunities for professional growth, and promoting a culture of self-starters with a vested interest in the value of their work. To do this, you need to collaborate with your staff rather than imposing top-down solutions. You can't promote engagement and self-organizing behavior by diktat; it has to be a grass-roots effort supported by leadership and organizational values.
