Have fun, looking forward to your report.
I'd like to know how it handles scenes with high dynamic range (I don't mean HDR though), a backlit portrait taken outside on a sunny day for example (thankfully we seem to have one in the UK today!).
Portraits with the subject cropped at waist height with a background which goes off into the distance to show the drop off in depth of field and bokeh. All the shallow depth of field shots on the DXO website are with the subject extremely close to the lens. With a small sensor I'd expect not a lot of focus drop off.
Studio scene with skin tones, also useful for checking sharpness.
Low light to see how the scene is rendered.
What are the raw files like to post process.
General opinion regarding usability of the device and the app.
Your comparison of the same scenes with the DXOOne, DSLR and iPhone is a good idea.
Hope you're able to shoot all of the above