Publicly available material on Intel Edison (both from Intel and, among other things, CES videos) would indicate Edison to be one step further towards the direction of embedded computing/automation, that is one step further that direction from Raspberry Pi. CPU horsepower of Edison is clearly lower than that if RPi, and RPi has very lowly CPU even by modern tablet/smartphone standards.
It is also likely system-on-chip implementation on Edison has no built-in graphics in the sense they're commonly understood. In most embedded applications, they're not necessary (and Intel didn't demo them!), and low number of contacts on the reverse side of the board (Secure Digital card pads + roughly dozen I/O pads, likely GPIO, PWM and/or SPI - and with small likelihood, USB?) makes prospect of RPi-like graphics unlikely.
Exact specifications of Intel Edison and its' SoC, apart from the fact it has Quartz based CPU and publicly announced peripherals, are unknown at this point. My strong personal hunch is that from Mathematica user viewpoint, Edison can be mostly considered a "headless" (that is, displayless, and largely UI-less) Linux platform running network-reachable Mathematica kernel with emphasis on access to whatever is connected to the I/O pins - with great likelihood simple GPIO-based on/off indicators and controls, PWM-based speed controls, and more complicated sensors, controllers and user interfaces on SPI.
Typically at tinkering phase, one would use Wolfram kernel on the board as a remote kernel from PC Mathematica front-end. I/O would be interactively accessible this way. Typical goal would be designing a control system or data logging and postprocessing device based on headless kernel and devices connected through the I/O. Once completed, code would be left to run - and start at boot - on the board, maybe with some level of "cloud" deployment - pushing data to and pulling it from some other network-accessible service. Wolfram Cloud, maybe? Limited UI (think of simplistic Manipulate) deployment through Bluetooth and a phone app would be... interesting, but maybe too much to hope.
If executed well, I can see Mma on Intel Edison (and hardware platforms that with almost certain likelihood will replace it in the future) synergize nicely with WRI cloud and connected devices efforts. There are missing pieces, though. One of the biggest in my opinion is licensing - is all this essentially limited to hobbyist endeavors? At this point, it'd seem so.