Being the primary on a good project is essential for your career. A good project is fundable, will yield multiple publications, instigate new areas of research, and give you something interesting to talk about both personally and professionally.

A good project has two key features: funding and interest.

Funding

Without money, nothing will get done. A fellowship can cover salary, but a project will suffer if there is no funding dedicated to it for supplies, travel, and other basics. The ideal postdoc or graduate student project should be related to a funded R01 grant, and of a scope and interest level to receive a fellowship.

Interest

It is not enough that you find your project interesting. Your project must be interesting to your PI, your department, your field, scientists in general, and the general public. Interesting projects are promoted. They're more likely to be funded, and easier to publish in high impact journals. Plus, it takes the same amount of time to complete a dull project as it does an interesting one. Why waste the time?

One thing many people downplay is the critical importance of having your PI’s interest in your project. PIs have a tendency to present only a subset of projects when asked to speak. They focus more energy on reading about topics they are fascinated by. You want to be on one of your PI’s favorite projects. You want them to be interested and attentive to your work. You want them to be asking you about data due to genuine curiosity.  You want their help in accessing resources. You want them to talk about your work to their friends and in professional settings. You want them to push for your publications.

It is also important that your field finds your project interesting. You want people to come to your poster or talk at a meeting. You want them to be curious about your results, to suggest potential collaborations expanding from it. You want people to request your reagents. You want them to ask you to visit to present your work. All these things will help you get published and help with your next career step. Also, your field is an easy audience. If they're not excited, it's not likely that people outside your field will be either. 

Independence

But here's where it can get tricky. Postdocs need something to take with them to start their own research group when they leave the lab. A PI will never let their favorite project go, and a newly made PI will never be able to compete with their former advisor. Think carefully about how you can take a project in a new direction. It can always be done, but it does require thought and planning.