I just thought I'd share where my postdoc fits in with some of the issues he raised.
1) Early vs. Later stage PI. Ruby-throated Hummingbird is early-ish in her career and hoping to go up for tenure next fall. That will depend partially on how much I can get done for her. As CPP pointed out, there is less competition to work with less famous peeps. In my case, none that I know of. I've been trying to cultivate this position for 2+ years and it was not publicly advertised. I did not take early vs. late career PI into account when searching for a postdoc. Geography and topic were the only major considerations. My PhD advisor was chosen partly on this criteria though.
2) The way the lab is organized or run. I was not hired to churn out data using a technique I have specialized in. I will need extensive training to be able to do my experiments, but will be able to bring my own point of view to the table. This particular lab is opposite the 'division of labor' model. There are a lot of experiments to be done using my specialized techniques, but I am only consulting on them while the grad students are working through them.
3) The subject of the research itself. To branch out in organism, technique or question. My organism is very similar to what I did my dissertation research on, and the questions I'm asking are related to theories I've been interested in since I started doing research. Mainly, it is the techniques and framework of questioning that are different.
EcoGeoFemme has also weighed in on the subject of really early career PIs. I agree with her point that there will be- or already are- an increasing number of people whose postdoc funding comes from a start-up package. This means you might not get a lot of the assumed benefits of a postdoc like access to a huge network. I think the change in Issue #1 should also lead to changes in issues #2 or 3. You have no way to know how the lab will be run. Seeing/helping a lab get set-up could be a good experience but probably takes a ton of time that could be better spent churning out data from a specific technique. Also, in a new lab, perhaps it is better to try to branch out more in your subject to try to increase your network as much as possible.
*Or, not so recently....but I get Internet installed at home on Monday! Yay!!!! Posting in a timely fashion!