OK buzzkills – no one ever said it would be cheap or easy.
As for traffic, you'll be surprised what a low memory optimised config can do.
Sorry – no-one wants to do that. But to avoid disappointment/wasting time and money, it is worth being vaguely realistic about how much power even a very low power (PC-type) device uses relative to the limited power (looked at as a 24/7 average in eg a UK winter) that even a sizeable solar panel generates.
What would be interesting (when possible) would be to get a real-life estimate of how much power a Pi operating in a minimal configuration (but still doing useful work) might consume. I guess since this idea involves a web server that it would need to be a Pi-B and so maybe the minimum might be eg 300mA, but there's no substitute for real data. And there's going to be some inefficiency supplying this from eg a 12v panel even with a good SM PSU, so you're probably looking at 2W continuous draw but it would be really interesting to know for sure. (As you might have guessed, I've got a first-hand interest in this – because this is how I'll be running a Pi.)
@Cracknel: I think you need to be thinking of at least a 30W panel and eg a 40-50Ah battery. Of course it all depends on how long you want the system to keep running if we get several days in succession of North Sea gloom (placed in a Cambridge, UK context), but even for non-critical systems, people often like to factor in a 10-day reserve.
And yes I agree, the wind turbine would be a nice supplement, but the cheapest 'serious' turbine I know of would be a Rutland 504 or similar where you're looking at eg £300 or thereabouts as the cost.
Overall, I'd suggest that the current draw for a Pi-type device needs to drop into the 20-50mA range (@5V) before it comes practicable to power it continuously from a relatively small/cheap solar PSU.