Update (Aug 1/2017): Uploaded the most recent "V3.0" design, replacing the STL/SCAD and pictures. Nudged things around slightly, and rotated the scoop holder 30 degrees so it doesn't hit the funnel anymore.
Here's my alternative take on Twiskck's and cingerick's Aeropress stand designs.
cingerick's design was too big to print on my printer. Twiskck's remix comes in 2 pieces so I could print it, and adds the scoop holder which is nice. But the two halves kept coming unglued with the crappy glue I was using, and I thought it needed handles.
I moved things around so everything fits in a 190x120 square, which can be printed in a single shot on most printers. I also added handles on the sides, which makes it easier to pick up.
I highly suggest printing this using PETG, as it can handle heat without warping and it's relatively food safe. Hand wash with a wet cloth.
OpenSCAD files are attached. Like most of my projects, it's a disorganized mess.