What is the absolute cheapest way to get a Raspberry Pi-style board, which has WIFI, and the ability to provide constant power using solar or other renewable means?

Or even just a really long-lasting battery rather than even renewable energy?

I am looking for links to products that I can buy

1. Board

2. WiFi

3. Power source

By the way, the WiFi chip must have the ability to run a relay... i.e. to be used to receive WiFi from a hotspot, and also to be used as a hotspot itself at the same time.

The device needs to do very little other than receive a WiFi connection and then repeat it in a relay-style, i.e. create a mobile hotspot.although not able to answer your question, reading your post made me think of a similar issue I was having with poor radio reception.

long story short, I was not able to acquire a signal, boost it and RETRANSMIT it, laws about then being CLASSED as a broadcaster or such and I would need a government license to do so.
so just check your state/country laws of doing what you are after first.Presently Solar power costs too much. And if the device has to run at night, you have to have a rechargeable battery.
You could use a 6 volt motorcycle battery. Use a power rectifier to drop the voltage to about 5.4 volts. A 5 amp hour battery cell is about $7 or less.
The Pi takes about 1/3 amp, so you could have MAYBE 15 hoers before you have to RECHARGE the battery.
If you still want solar, you have to do better hand 1/3 amp, more like almost 1 amp at 6 volts. Because you have to charge the battery, run the PI and have some overhead because the Sun is only powerful for a few hours. So the solar panels might cost you more that the Pi. Maybe 415 to $20.

Have you already seen this?
Setting up a Raspberry Pi as a WiFi access pointWhat is your full plan for the device? How many users is it going to serve? Are you trying to create some sort of point to point network between two buildings?