More than anything, labs want to be sure of a few things before they set you free on your own project:
1) They want to know that you are competent with the basic lab skills your project would require. Because of this, many labs want lab newbies to help out with the technical aspects of existing projects before giving them too much freedom.
2) They want to know that you understand the process of the lab's research. This often means sitting in on lab meetings even when you have nothing to present, or having theoretical talks with the PI about logical ways to keep existing projects moving. Convincing a PI that you are competent on this front is usually much harder than convincing them that you can handle the manual skills - here, the more proactive you are about sitting in on meetings, taking additional roles in existing projects, etc. the better.
3) They want to know that what you want to do will in some way help their research goals. This means that you are more likely to end up with your own project if you suggest something that is an expansion of the work the lab is already doing. Most of the projects I have handled in the past have been the result of saying, "Hey, have you looked into related thing x, y, or z yet? Would you like me to?". Again, it's about being proactive.
If you are just joining a lab, be sure to fill in the PI on any past experience you have had (a good, thorough CV is a great way to do this). Also, you should probably ask (either directly or indirectly) about the lab's funding - if funding is low, you probably won't end up with your own project regardless, simply because there isn't enough money to throw at you. If you are already in a lab that seems to be moving in a way you aren't happy with, try to work on what I have said above, it usually helps.