PS VITA 2000
Personally, I really like the feel of this stick. Instead of the slide-around motion of the PSP stick, it actually tilts like a typical analog stick. The range of motion feels a bit small, but the tilt motion feels much more comfortable to play with. A downside to this is that you need a 0.5mm 6-pin connector in order to make it work, but those are cheap so it's not a huge issue.
The stick can be found on Aliexpress for $3.
I got the connector from Walmart because I couldn't wait a month.
This pinout is specific to this orientation which I felt would fit best in the GBZ, but if you rotate it all you need to do is swap the X and Y connections.
This is probably the most common stick you'll see in builds. The base has more depth than the Vita stick, but it still fits perfectly between the shell and any AIO PCB you may have. Kite recommended it for the SAIO build, and the pinout image is from Kite's SAIO support page. One thing to note about this one is that it has a dust protector around the base of the stick, which when pressed against the GBZ shell can cause a lot of stiffness/friction when trying to rotate the stick. I just popped the joystick's grip off and removed that piece, and now it works like a charm. This stick also seems to be the easiest to solder n' go, with 4 large pads on the bottom.