You can run a fair bit of web traffic off of something like a Raspberry Pi, and you don't need a ton of battery to keep that running overnight. Heavy database driven websites probably won't be an option, but the bottleneck for static sites would likely be the Ethernet interface.

In fact that's pretty much exactly what they did. 168Wh battery pack is a small Deep Cycle SLA. A 50w Solar Panel and associated charge controllers and the like is like $80 at Harbor Freight. The whole thing is quite achievable on a budget.

I tried running an rpi on a lead acid battery and a 40w panel. It seemed to be running fine for the first 2 weeks but then I think it had a few days where it drained the battery to 0 which ruined it and then it was turning off every night. I'm not sure what to do with the setup now since it seems like lead acid is not the way to go but all of the DIY solar charge controllers use lead acid.

Did you use a deep cycle battery? Letting a standard lead acid battery go to 0 is a surefire way to kill it.

It was just a standard one but the deep cycle ones are super expensive and it doesn't seem worth it for a rpi that costs about $1 a year to run.

As is often the case, the most obvious target for optimization (the computer, 1-2W) has already hit diminishing returns. Their biggest energy hog is the router (10W), which they aren't running on solar.

but what's the manufacturing process of router chips ? surely a router inner logic is way less than a rpi SoC so it could be trimmed down over time through simple smaller features .. ?

Home routers are just a general purpose CPU (ARM or MIPS) running a trimmed down OS. Thats why you can install openWRT on a lot of them and run arbitrary code.

IMO the problem is like with cable boxes- forget about the manufacturing node, there's just little incentive for the router companies to optimize power because few people pay attention to it.