this has been done with a lot of embedded type devices already, look up about e.g. mesh potato (low power telecom / wireless mesh bubble) - I love building remote/standalone stations with those things - http://developer.servalproject.org/dokuwiki/doku.php?id=content:meshextender:solar_operation
(you can see it already gets massive, they are using 12V 30aH battery and 40W solar panel, and meshpotato is not a device that uses much power)
it's really just all arithmetic...
- how many hours per day will the edison need to be on battery
- how much power does it draw per hour
--> gives you what amp-hour battery you need
(and you probably really don't want to be draining the batter full-dead overnight every time - you really don't want it getting less than 50% or maybe even not that low)
- typical retail panels usually end up at about 17-18V in full sun (barely squeaking by if directly connected to J21, and I have no idea if it would be tolerated if the panel happened to spike at say 19V or 20V - open-circuit they can max more like 22-23V, even.)
(so you might want a charge controller simply to drop the "load" (aka edison) side down to 12V. inexpensive controllers are rated as low as 5A, 7A, 10A. also if you use a charge controller, you could completely skip the edison J2 minimalist battery specs, and connect a big honking SLA/glass-mat 12V battery to the charge controller itself)
if you are building your own panel from components, obvs you have direct control over the output voltage, plus you could put DC regulation etc.)
- what actual wattage output do you need to:
-- a. power the edison? (obvs different given how it's being used, etc.)
-- b. charge the battery at a fast enough rate to recover from the overnight loss
an inexpensive 20w panel might give a bit over 1A at best component quality, and the output voltage is what actually varies. if you are getting a lot of sun, 1A @ 18V will give you 18 watts. poor sun maybe 11-12 watts or less. naturally there are losses around every corner, too.
do the math, e.g. if your edison only ever uses, say, 800ma @ 5V (call it 400ma @12-13V), does the remaining power from the solar provide enough charge to top off your battery. etc. etc. (and does it provide it fast enough over the hours the sun is shining)
and of course start to increase all specs because your geography will factor in how many hours of sun you get, how useful each hour of sun actually is, etc.